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1 TMA41/MMA4 Functional Analysis 21/211 Peter Kumlin Mathematics Chalmers & GU Gamla tentor från 2 27 lösningsförslag levereras separat 1

2 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon:Magnus G Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Set f 1 (x) = 1, f 2 (x) = x and f 3 (x) = x 2. Find an ON-sequence (g n ) 3 n=1 such that Span{f n : n = 1, 2, 3} = Span{g n : n = 1, 2, 3} in the Hilbert space of realvalued funtions on [.1] with the inner product h 1, h 2 = h 1 (x)h 2 (x)xdx. 2. Show that the boundary value problem { u (x) + u (x) + sin u(x) = cos x, x 1 u() =, u(1) = has a unique solution u C Let X be a Banach space and let T : X X be a mapping with the following property: There exist real numbers α > 1 and C > such that T(x) T(y) C x y α for all x, y X. Show that there exists a x X such that T(x) = x for all x X. 4. Give the definition of compact operator on a Hilbert space and show that every compact operator is bounded. 2

3 5. Give the definition of strong and weak convergence of a sequence on a Hilbert space H. Show that strong convergence implies weak convergence. Give an example of a weakly convergent sequence on a Hilbert space that does not converge strongly (including a proof of the statement). (5p) 6. Assume that T : X X is a mapping on a Banach space X and that there exist real numbers c n, n = 1, 2, 3,... satisfying Σ n=1c n < such that T n x T n y c n x y for all positive integers n and all x, y X. Show that the sequence (T n z) n=1 converges for every z X and that the limit is a fixed point for T and that the fixed point is unique. Good Luck!! PK 3

4 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Elisabeth Wulcan Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Set Tf(x) = 2π sin(x t)f(t) dt, x [, 2π]. Show that T is a bounded linear and compact operator when T is considered as an operator on (a) the Banach space C([, 2π]), (b) the Banach space L 2 ([, 2π]). 2. Show that the boundary value problem { u (x) + λ sin u(x 2 ) = u(x), x 1 u() =, u(1) = has a unique solution u C 2 for λ small enough. Give an estimate λ > such that the problem has a unique solution for all λ < λ. 3. Let (e n ) n=1 be an ON basis for a Hilbert space H. For each n set f n = e n+1 e n. Show that span{f n : n = 1, 2, 3,...} is dense in H. 4. State and prove (a version of) Banach s contraction fixed point theorem. 4

5 5. Let A be a bounded linear operator H H where H is a Hilbert space. Define the adjoint operator A and show that it is a well-defined bounded linear mapping on H with A = A. (5p) 6. We say that an inner product space (E,, ) has the Riesz property if every bounded linear functional f on E is given by f(x) = x, x for some x E. Riesz representation theorem implies that every Hilbert space E has the Riesz property. Show that every inner product space with the Riesz property must be a Hilbert space. Good Luck!! PK 5

6 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Aron L/Roger A Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Show that there exists a unique C 2 -function u(x) defined on [, 1] with u() = u(1) = such that u (x) cos 2 u(x) = 1, x [, 1]. 2. Show that there exists a function f(x) defined on [, 1] with f(x) 2 dx = 1 such that f(x)e x dx = sup{ Calculate f(x) and f(x)e x dx. g(x)e x dx : g(x) 2 dx = 1}. 3. Suppose f : Ê Ê satisfies f(x) f(y) < x y whenever x y and has no fixed points. Show that either or f n (x) + as n for all x Ê f n (x) as n for all x Ê. Here f n denotes the composition of f with itself n times. 4. State and prove the Riesz representation theorem. (5p) 6

7 5. Let A be a bounded linear operator on a Hilbert space H. Define the adjoint operator A, show that it exists as a bounded linear operator on H and show that A = A. 6. Let T be a linear mapping on a complex inner product space E such that Tx = x for all x E. Show that Tx, Ty = x, y for all x, y E. Good Luck!! PK 7

8 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Oscar Marmon Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Show that the family of all solutions of the ODE y = y in the interval (, 1) is a subspace of C((, 1)). Show that the family of all solutions of y = y 2 is not a subspace of C((, 1)). 2. Show that Y = {Ü = (x n ) n= l2 : x 2k =, k =,1,2,...} is a closed subspace of l 2 and find Y. 3. Consider f(x) + c (x + t)f(t) dt = g(x) C([, 1]), x [, 1]. Assuming c c 12, solve the equation. 4. State and prove the Lax-Milgram theorem. 5. Show that the adjoint operator of a compact operator is compact. (5p) 6. Suppose 1 f : Ê Ê satisfies f(x) f(y) < x y whenever x y. Show that there exists a ξ [, ] such that for any real x, f n (x) ξ as n. Here f n denotes the composition of f with itself n times. 1 Hint: Consider the the two cases that f has a fixed point and that f has no fixed point respectively. 8

9 Good Luck!! PK 9

10 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Milena Angelova , Peter Kumlin Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Consider the complex vector space c of all sequences Ü = (x n ) n=1 such that x n in as n. Let (x n) n=1 be the decreasing rearrangement 2 of ( x n ) n=1. Define for Ü = (x n) n=1 c Ü = sup m=1,2,3,... 1 m Σ m n=1 x n and set d = {Ü c : Ü < }. Show that is a norm on d. 2. Show that the boundary value problem { u (x) + u 2 (x 2 ) =, x 1 u() =, u(1) =, u C 2 ([, 1]) has a solution u with the property max x 1 u(x) 1. Show that the solution is unique. 3. Let T be the integral operator on the complex Hilbert space L 2 ([ π, π]) defined by Tf(x) = π π k(x, t)f(t) dt, 2 For each positive integer n we denote by x n the real number x that satisfies {k : x k > x} < n {k : x k x}. Here A denotes the number of elements in the set A. 1

11 where k(x, t) = (sin x + sin t) 2 1. Show that T is self-adjoint and calculate 8 T. Find all nonzero eigenvalues and corresponding eigenfunctions for T and determine σ(t). Solve the equation in L 2 ([ π, π]). Tu = πu 5π 4 4. State and prove the Riesz representation theorem. 5. Let (x n ) n=1 be a weakly convergent sequence in a Hilbert space H. Prove that (x n ) n=1 is a bounded sequence in H. (5p) 6. Let T : H H be a compact linear mapping on a Hilbert space H. Show that (a) dim N(I + T) < (b) dim N(I + T ) = dim N(I + T) Here I denotes the identity operator on H and T the adjoint operator of T. Good Luck!! PK 11

12 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 3,4,5. Telefon: Marcus Better, Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Prove the existence and uniqueness of a solution to the following boundary value problem: 2. Set H = L 2 ([, 1]). Let T be given by Show that u 1 (x) = u 2 (x), x 1 u() = u(1) =, u C 2 ([, 1]). Tf(x) = 1 x f(t) dt. (a) T is a bounded linear operator on H and (b) calculate the kernel k(x, t) for T and show that T is self-adjoint. (c) Moreover calculate the kernel k 2 (x, t) for T 2 and (d) find 3 all eigenvalues and eigenfunctions for T. (e) Finally calculate T. (8p) 3 Hint: Let f be an eigenfunction for T and calculate (T 2 f). Show that f is a solution to the equation λ 2 f + f =. 12

13 3. State and prove the Lax-Milgram Theorem. (5p) 4. Let K(H) be the subset of all compact linear operators H H on a Hilbert space H in B(H) with the operator norm. Show that K(H) is a closed subspace in B(H). 5. For non-negative integers k let C k (, 1) be the vector space of k times continuously differentiable functions f : (, 1) Ê such that f k = sup Σ k l= f(l) (t) <. t (,1) Show that X k = (C k (, 1), k ) is a Banach space and that the identity map I : X k X k 1, f f, is a compact operator. Good Luck!! PK 13

14 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Johan Jansson, Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Show that the boundary value problem { u (x) + u(x) + λ cos(1 + u(x)) =, x 1 u() =, u () = 1, u C 2 ([, 1]) has a unique solution for λ ǫ, ǫ small. Give an upper bound on ǫ. 2. Define T : L 2 ([, 1]) L 2 ([, 1]) by Tf(x) = k(x, t)f(t) dt, where Find σ(t). k(x, t) = { 1 x t x < t. 3. Let x n = (,,...,, 1, 2,,...) where the numbers 1 and 2 appear in the positions n and n + 1 and let y n = (1, 1,..., 1,,,...) with the number 1 in the first n positions. Consider these as vectors in l 2. Prove that for all n = 1, 2,... y n Span{x 1, x 2,...}. 14

15 4. State and prove the Lax-Milgram Theorem. 5. Show that every weakly convergent sequence in a Hilbert space is bounded. (5p) 6. Let f : E be a linear functional on a normed space E. Assume that N(f) is closed. Show that f is bounded. Good Luck!! PK 15

16 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Peter Kumlin , Lärare besöker skrivsalen ca 9.3 och Datum: Skrivtid: fm (5 timmar) Lokal: V 1. Show that the boundary value problem { u (x) + u(x) + λ cos(1 + u(x)) =, x 1 u() = u () =, u C 2 ([, 1]) has a unique solution for λ ǫ, ǫ small. Give an upper bound on ǫ. 2. Let (e n ) n=1 be an ON-basis in a Hilbert space H and define the operator T by T(Σ n=1 a ne n ) = Σ 1 n=2 n a ne n 1. Show that T is compact and find T. Find 4 σ p (T) and σ p (T ). 3. Let f 1, f 2,..., f n B(H, ) be linearly independent where H is a Hilbert space. Show that there exist x 1, x 2,...,x n H such that { 1 i = j f i (x j ) = i j for all i = 1, 2,..., n. 4 σ p (A) = {λ : λ eigenvalue to A}. 16

17 4. State and prove the Orthogonal Projection theorem 5. Also the Closest Point Property theorem should be proved. 5. Let T B(X, X) where X is a Banach space and T < 1. Show that I + T is an invertible operator, i.e. (I + T) 1 B(X, X). (5p) 6. Let T : H H be a linear mapping in a Hilbert space H. Assume that Tx n Tx for every x n x. Show that T B(H, H). Good Luck!! PK 5 Often referred to as the Orthogonal Decomposition theorem. 17

18 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Datum: Skrivtid: fm (5 timmar) 1. Show that the following boundary value problem { u (x) + u (x) = arctan u(x 2 ), x 1 u() = u(1) =, u C 2 ([, 1]) has a unique solution. 2. Let T be a positive compact self-adjoint operator on a Hilbert space H with T 1. Give an upper estimate 6 for 3T 4 2T 3 + T Let k L 2 ([, π] [, π]) and consider the linear mapping given by Tf(x) = T : L 2 ([, π]) L 2 ([, π]) π k(x, y)f(y) dy, x [, π] for f L 2 ([, π]). One standard estimate for the operator norm for T is T k L 2 ([,π] [,π]). Prove 7 that also the following estimate is true: T (sup x π k(x, y) dy) 1 2 (sup y π k(x, y) dx) 1 2. Finally apply these two estimates to the kernel function k(x, y) = cos(x y), i.e. calculate the two upper bounds for the operator norm. 6 Better than the trivial estimate 24 7 Apply the formula g = sup h =1 g, h to Tf. Also the estimate ab c 2 a c b2 for all a, b Ê and c > can come in handy. 18

19 4. State and prove Banach s fixed point theorem. (5p) 5. State and prove the orthogonal projection theorem. 6. Let X be a finite-dimensional vector space with a norm. Moreover let T be a linear mapping on X that is 1 1. Show that there exists a C > such that Tx C x all x X. Good Luck!! PK 19

20 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Rolf Liljendahl Datum: Skrivtid: fm (5 timmar) 1. Show that the following boundary value problem 5u 1 (x) u(x) = 1, x 1 4 u() = u(1) =, u C 2 ([, 1]) has a unique solution. 2. Let (e n ) n=1 be an orthonormal basis for a Hilbert space H and set f = e 1 f k = e 2k+1 k >. f k = e 2k k < Moreover define S by S(Σ k= a kf k ) = Σ k= a kf k+1. Show that S is a welldefined bounded linear mapping on H, calculate S and show that S has no eigenvalues. 3. Let (e k ) n k=1 be a sequence of vectors in a Hilbert space H. Assume that e k = 1 for all k. Show 8 that Σ n k=1 x, e k 2 x 2 (1 + (Σ n k,l=1 e k, e l 2 ) 1 2 ) k l for all x H. Note that if (e k ) n k=1 is an ON-sequence in H then the statement is called Bessel s inequality. 8 Hint: Note that Σ x, e k 2 = x, Σ x, e k e k. 2

21 4. State and prove the Lax-Milgram theorem. (5p) 5. Let P, Q be orthogonal projections on a Hilbert space such that PQ = QP. Show that P + Q PQ is an orthogonal projection. 6. Let T : H H be a compact linear operator on a Hilbert space H. Show that R(I + T) is a closed subspace of H. Good Luck!! PK 21

22 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Peter Kumlin eller Datum: Skrivtid: fm (5 timmar) 1. Show that the following boundary value problem { u (x) u(x) (1 + u(x2 )) =, x 1 has a unique solution. u() = u () =, u C 2 ([, 1]) 2. Let H = L 2 ([a, b]), a, b finite, and Tf(x) = 1 b a b a f(x) dx, x [a, b]. Show that T is a bounded linear operator H H and that T is a projection. 3. Let h C([, 1] [, 1]) be a real-valued function such that for all x, y [, 1]. Set Tf(x) = h(x, y) = h(y, x) > h(x, y)f(y) dy, x [, 1] for f L 2 ([, 1]). Show that the bounded linear operator T on L 2 ([, 1]) has an eigenvalue λ = T which is simple. 22

23 4. State and prove the Orthogonal Projection theorem 9. Also the Closest Point Property theorem should be proved. (5p) 5. Define the notion of weak convergence on a Hilbert space and show that every weakly convergent sequence is bounded. 6. Show that for every compact self-adjoint operator T on a Hilbert space there exists an eigenvalue λ of T with λ = T. Also show that there can be no eigenvalue µ of T with µ > T. Good Luck!! PK 9 Often referred to as the Orthogonal Decomposition theorem. 23

24 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Karin Kraft Datum: Skrivtid: fm (5 timmar) 1. Show that the following boundary value problem u (x) + u(x) = u(x) 2 + u 2 (x), x π 2 u() = u( π) =, u 2 C2 ([, π]) 2 has a unique solution. 2. Let T be the linear mapping on L 2 ([, 1]) defined by Tf(x) = Show that T is bounded and calculate T. (x + y)f(y) dy, x Show that the following boundary value problem (almost the same as problem 1) u u(x) (x) + u(x) = λ 2 + u 2 (x), x π 2 u() = u( π) =, u 2 C2 ([, π]) 2 has a solution for all λ Ê. 24

25 4. State and prove the Riesz representation theorem. (5p) 5. Let T be a mapping on a real normed space X satisfying T(x + y) = T(x) + T(y) for all x, y X. Show that T(λx) = λt(x) for all λ R and x X if T is continuous. 6. Let (x n ) n=1 be a complete ON-sequence in a Hilbert space H and let (y n) n=1 be another ON-sequence such that Σ n=1 x n y n 2 < 1. Show that the ON-sequence (y n ) n=1 also is complete. Good Luck!! PK 25

26 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Anders Logg Datum: Skrivtid: fm (5 timmar) 1. Show that the following boundary value problem u u(x) (x) u(x) = λ 1 + u 2 (x), x 1 u() u () + u(1) = u() + u () + 2u (1) =, u C 2 ([, 1]) has a unique solution for λ < ǫ where ǫ > is close to. Give an estimate on the size of ǫ. 2. Show that the set {f 1 (x), f 2 (x), f 3 (x),...}, where f n (x) = x n, 1 x 1 and n = 1, 2, 3...., is linearly independent and use the Gram-Schmidt process (with the Hilbert space L 2 ([ 1, 1])) to produce the first three orthogonal vectors, call them g 1, g 2, g 3, out of f 1, f 2, f Let (u n ) n=1 be an othonormal sequence in L 2 ([, 1]). Show that the sequence is an orthonormal basis if x Σ n=1 u n (t) dt 2 = x, for all x [, 1]. 26

27 4. State and prove the Riesz representation theorem. (5p) 5. Let H be a Hilbert space. Prove or disprove the statement: Every bounded linear mapping on H preserves orthogonality. 6. Let X be a separable Hilbert space and T : X X a compact linear operator. Show that T can be approximated by finite rank operators in B(H), i.e. there exist a sequence of finite rank operators T n on H such that T n T in operator norm. Good Luck!! PK 27

28 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Axel Målqvist Datum: Skrivtid: fm (5 timmar) 1. Prove the existence and uniqueness of a solution to the following boundary value problem { u (x) + u (x) = arctan u(x 2 ), x 1 u() = u(1) =, u C 2 ([, 1]) 2. Let (e n ) n=1 be an ON-basis for a Hilbert space H and assume that T : H H is a bounded linear operator on H such that Σ n=1 Te n 2 <. Show that if (f n ) n=1 is another ON-basis for H then Σ n=1 Tf n 2 = Σ n=1 Te n 2. Moreover show that T 2 Σ n=1 Te n Set Ê+ = {x Ê : x }. For f L 2 (Ê+) define Show that Mf(x) = 1 x x f(t) dt, x >. M : L 2 (Ê+) L 2 (Ê+) is a bounded linear mapping on L 2 (Ê+), calculate the operator norm of I M and, finally, determine the adjoint operator of M. Here I denotes the identity operator on L 2 (Ê+). 28

29 4. State and prove the Riesz representation theorem. (5p) 5. Let K(H) be the subset of all compact linear operators H H on a Hilbert space H in B(H) with the operator norm. Show that K(H) is a closed subspace in B(H). 6. Let X be a Banach space and T : X X a compact 1 linear operator. Show that there exists a constant C such that for every y R(I + T) there exists a x X with y = (I + T)x such that x C y. Good Luck!! PK 1 Exactly the same definition as for a linear operator on a Hilbert space 29

30 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Erik Broman Datum: Skrivtid: fm (5 timmar) Lokal: VV22 1. Let A be the linear mapping on L 2 ([, 1]) defined by Calculate Af(x) = (x y) 2 f(y) dy, x 1. (a) A (b) A. (1+3p) 2. Consider the differential equation { u = λe u, < x < 1, u() = u(1) =. (a) Formulate the boundary value problem as a fixed point problem u = Tu, where T is an integral operator. (b) Set B = {u C([, 1]) : u 1}. Show that T maps B into itself provided < λ < λ for λ sufficiently small. Give a numerical value on λ. (c) Show that the differential equation is uniquely solvable in B with λ chosen as in (b). (2+1+1p) 3. Let T be a positive, self-adjoint, compact operator on a Hilbert space H. Show that Tx, x n T n x, x x, x 2(n 1), for all positive integers n and all x H. 3

31 4. State and prove Lax-Milgram s theorem. (5p) 5. State and prove the orthogonal projection theorem. 6. Let A be a subset of a Hilbert space H. Show that (A ) is the smallest closed subspace of H that contains A. Good Luck!! PK 31

32 Matematik, CTH & GU Tentamen i Funktionalanalys TMA41/MAN67 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Peter Kumlin (eller ) (Om telefonen ovan ej fungerar: Jana Madjarova ) Datum: Skrivtid: fm (5 timmar) Lokal: maskin 1. Let A be the linear mapping on L 2 ([, 1]) defined by Calculate Af(x) = (x y)f(y) dy, x 1. (a) A A (b) A. (2+2p) 2. Prove the existence and uniqueness of a solution to the following boundary value problem u 1 (x) = u 2 (x), x 1 u() = u(1) =, u C 2 ([, 1]) 3. Let T : X X be a mapping (not necessary linear) on a normed space X. Moreover assume that there are real constants C, α, where α > 1, such that T(x) T(y) C x y α, for all x, y X. Show that there exists a z X such that T(x) = z for all x X. 32

33 4. State and prove Hilbert-Schmidt s theorem. (5p) 5. Let A be a bounded operator on a Hilbert space H. Define the adjoint operator A (also prove that it exists) and show that A is a bounded operator on H with A = A. 6. Let T be a self-adjoint operator on a Hilbert space H. Assume that T n is compact for some integer n 2. Prove that T is compact. Good Luck!! PK 33

34 Matematik, CTH Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Peter Kumlin (alternativt ) Datum: Skrivtid: fm (5 timmar) Lokal: maskin 1. Låt H vara ett oändligtdimensionellt Hilbertrum och T : H C en begränsad linjär funktional. Beräkna dimensionen för det linjära delrummet N(T) av H. Ge också ett exempel på ett Hilbertrum H och en funktional T som ovan. 2. Visa att det finns exakt en två gånger kontinuerligt deriverbar funktion u(x) definierad på intervallet [, 1] sådan att u() = u(1) = och u (x) cos 2 u(x) = 1, x [, 1]. 3. Låt T vara en självadjungerad, positiv, kompakt operator på ett Hilbertrum H med T 1. Ge en uppskattning 11 av 3T 4 2T 3 + T Formulera och bevisa Lax-Milgrams sats. (5p) 5. Låt T vara en begränsad linjär operator på ett Banachrum X. Definiera σ(t). Låt σ a (T) beteckna det approximativa spektrumet för T, dvs mängden av alla komplexa tal λ för vilka det finns en följder {x n } n=1 i X med x n = 1 så att lim (T λi)x n =. n Visa att σ a (T) är en delmängd av σ(t). 11 En uppskattning bättre än den triviala 3T 4 2T 3 + T 2 3 T T 3 + T

35 6. Givet en tät delmängd S i ett Banachrum X. Låt vidare {T n } n=1 vara en följd av linjära operatorer på X. Antag att (a) lim n T n x existerar för alla x S och att (b) det finns ett C > så att för alla n och alla x X. T n x C x Visa att lim n T n x existerar för alla x X. Motivera väl! Lycka till!! PK 35

36 Matematik, CTH Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Robert Berman Datum: Skrivtid: em (5 timmar) Lokal: 1. Låt l 2 beteckna Banachrummet av alla sekvenser (...,x 2, x 1, x, x 1, x 2,...) med elementvis addition och multiplikation med skalär och med den vanliga normen x 2 = Σ n= x n 2. För varje x l 2 definiera { xn+1 + 2x (Tx) n = n 1 + 1x n, n = 2k, k Z 2x n+1 + x n 1 + 1x n, n = 2k + 1, k Z. Avgör om T är (a) en begränsad linjär operator på l 2 (b) självadjungerad (c) en inverterbar operator Visa att det finns exakt en två gånger kontinuerligt deriverbar funktion u(x) definierad på intervallet [, 1] sådan att och u() 2u(1) = u () 2u (1) = 4u (x) u(x) + x =, x [, 1]. 3. Låt T vara en begränsad linjär operator på ett Hilbertrum H med dim R(T) = 1. Visa att för alla y R(T), y, finns entydigt bestämda x H så att Visa också att Tz = z, x y, z H. T = x y. Använd t.ex. detta faktum för att beräkna operatornormen för avbildningen 12 dvs T 1 B(l 2 ). Tf(t) = e t s f(s) ds, f L 2 [, 1]. 36

37 4. Formulera 13 och bevisa Banachs fixpunktssats. 5. Formulera och bevisa Riesz representationssats. (5p) 6. Givet ett slutet äkta delrum F i ett normerat rum E. Visa att det för varje ǫ > finns ett x E sådant att x = 1 och x y > 1 ǫ för alla y F. Använd t.ex. detta för att visa varje normerat rum X där den slutna enhetsbollen {x X : x 1} i X är kompakt är ändligtdimensionellt. Motivera väl! Lycka till!! PK 13 Antingen den version som finns i kursboken eller den som är given i fixpunktshäftet. 37

38 Matematik, CTH Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Datum: Skrivtid: (5 timmar) Lokal: VV 1. Sätt Au(x) = π e x+y cos(x + y)u(y) dy, x [, π]. Beräkna operatornormen för A och avgör om A är en kompakt operator då A betraktas som en operator på (a) Banachrummet C[, π], (b) Banachrummet L 2 [, π]. (2p+2p) 2. Visa att det finns exakt en två gånger kontinuerligt deriverbar funktion u(x) definierad på intervallet [, 1] sådan att u() = u () = och u (x) u(x) (1 + u(x2 )) =, x [, 1]. 3. Låt E vara ett normerat rum. Visa att det inte kan finnas avbildningar S, T B(E) sådana att ST TS = I, där I betecknar identitetsoperatorn på E. 38

39 4. Formulera 14 och bevisa Banachs fixpunktssats. 5. Formulera och bevisa 15 spektralsatsen för självadjungerade kompakta operatorer på Hilbertrum. (5p) 6. Låt T vara en normal linjär avbildning på ett Hilbertrum H, dvs T är en begränsad operator sådan att T kommuterar med sin adjunkt T, eller i klartext Visa att (a) Tx = T x för alla x H; TT = T T. (b) λ är ett egenvärde med egenvektor x till T om och endast om λ egenvärde med egenvektor x till T. (1p+3p) Motivera väl! Lycka till!! PK 14 Antingen den version som finns i kursboken eller den som är given i fixpunktshäftet. 15 Beviset ska inkludera bevis av den sats som kallas för Hilbert-Schmidts sats i kursboken. 39

40 Matematik, CTH Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon:... Datum: Skrivtid: (5 timmar) Lokal: Maskinhuset 1. För u C[, 1] sätt (Au)(x) = x x y u(y) dy, x [, 1]. Visa att A är en begränsad linjär operator på Banachrummet C[, 1] samt beräkna operatornormen A. 2. Visa att det finns exakt en två gånger kontinuerligt deriverbar funktion u(x) definierad i intervallet [, π 2 ] sådan att u () = u ( π 2 ) = och u (x) + u(x) = 1 2 sin u(x2 ), x [, π 2 ]. Beräkna här först Greenfunktionen och formulera sedan om differentialekvationen som en integralekvation. Bestäm slutligen funktionen u(x). 3. Antag att H är ett Hilbertrum. Använd spektralsatsen för att finna en H-värd lösning u(t) till begynnelsevärdesproblemet du (t) + Au(t) =, t >, dt u() = u H, där A är en kompakt, självadjungerad, positivt definit operator på H. Visa att u(t) u, t. 4

41 4. (a) Låt A vara en begränsad linjär operator på ett Hilbertrum H. Visa att N(A ) = R(A). (b) Definiera vad som menas med att en följd i ett Hilbertrum konvergerar svagt. Ge exempel på en följd som konvergerar svagt men ej starkt. 5. Formulera och bevisa Riesz representationssats. (5p) 6. Låt H vara ett komplext Hilbertrum och A en begränsad linjär operator på H med egenskapen att Ax, x R för alla x H. Visa att A är självadjungerad. Motivera väl! Lycka till!! PK 41

42 Matematik, CTH Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Niklas Lindholm, Datum: Skrivtid: (5 timmar) Lokal: VV 1. Låt (a n ) n=1 vara en begränsad följd, dvs (a n) n=1 l. Visa genom att använda Banachs kontraktionssats 16 att det finns en begränsad följd (x n ) n=1 som löser x n 1 + 4x n + x n+1 = a n, n = 1, 2,..., där x = Beräkna normen av operatorn A : C[, π] C[, π] given av (Af)(x) = π (1 + e i(x y) )f(y) dy. Beräkna också normen av operatorn B : L 2 [, π] L 2 [, π] given av (Bf)(x) = Funktionerna är komplexvärda. π 3. Låt T vara definierad för x = (x n ) n=1 enligt (1 + e i(x y) )f(y) dy. (Tx) n = nx n, n = 1, 2,... Visa att D(T) = {x l 2 : Tx l 2 } är en tät delmängd i l 2 och att T är en sluten operator 17 i l 2, dvs att x n l 2 för n = 1, 2,..., x n y i l 2, Tx n z i l 2 medför att y D(T) och Ty = z. 16 Betrakta avbildningen x n 1 4 (a n x n 1 x n+1 ), n = 1, Utnyttja t.ex. att T är en självadjungerad operator. 42

43 4. Låt c, c 1,...,c n 1 vara kontinuerliga funktioner på intervallet I = [, 1], där n är ett heltal 2. Låt vidare α ij, β ij för i =,...,n 1 och j = 1,..., n vara komplexa tal och sätt Vidare sätt och R j u = Σ n 1 i= (α iju (i) () + β ij u (i) (1)), j = 1,..., n. Lu = u (n) + c n 1 u (n 1) c 1 u (1) + c u Ru = (R 1 u,..., R n u). Ge tillräckliga villkor för att det för varje f C(I) ska finnas en unik lösning u C n (I) till problemet { Lu = f Ru =. Redogör dessutom för hur man beräknar u, dvs beskriv hur man bestämmer Greenfunktionen till randvärdesproblemet. 5. Formulera och bevisa Orthogonal decomposition theorem. (5p) 6. Låt H vara ett komplext Hilbertrum och A en begränsad linjär operator på H med egenskapen att Ax, x R för alla x H. Visa att A är självadjungerad. Motivera väl! Lycka till!! PK 43

44 Matematik, CTH Extra Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Datum: Skrivtid: em (5 timmar) Lokal: 1. Beräkna normen av operatorn A : C[, π] C[, π] given av (Af)(x) = π (1 + e i(x y) )f(y) dy. Beräkna också normen av operatorn B : L 2 [, π] L 2 [, π] given av (Bf)(x) = π (1 + e i(x y) )f(y) dy. 2. Antag att f C([, 1]) och λ R. Visa att ekvationen { u (x) + u (x) + λ u(x) = f(x), x [, 1] u() = u(1) =, u C 2 ([, 1]) har en entydigt bestämd lösning om λ är tillräckligt litet. 3. Antag att T : L 2 [, 1] L 2 [, 1] är en linjär avbildning sådan att Tf om f. Visa att T är en begränsad operator. 4. Definiera vad som menas med (ortogonal) projektionsoperator, att en operator är idempotent, samt visa följande påstående: Antag att A är en begränsad linjär operator på ett Hilbertrum H. Visa att A är en (ortogonal) projektionsoperator på H om och endast om A är idempotent och självadjungerad. 44

45 5. Formulera och bevisa Riesz representationssats. (5p) 6. Låt H vara ett komplext Hilbertrum och A en begränsad linjär operator på H med egenskapen att Ax, x R för alla x H. Visa att A är självadjungerad. Motivera väl! Lycka till!! PK 45

46 Matematik, CTH Tentamen i Funktionalanalys TMA4 Hjälpmedel: Inga (inte ens räknedosa). Personuppgifter: Namn, personnummer, linje, antagningsår. Inlämning ska ske i uppgifternas ordning; v.g. sidnumrera! Teoriuppgifter: 4,5,6. Telefon: Peter Kumlin Datum: Skrivtid: (5 timmar) Lokal: Gamla M-huset 1. Betrakta integraloperatorn Af(x) = 2π cos(x y)f(y) dy, x 2π. Visa att A definierar en begränsad linjär operator på de två Banachrummen (reella funktioner) (a) C[, 2π] (b) L 2 [, 2π]. Beräkna operatornormen A i något av fallen. 2. Antag att f C([, 1]) och λ R. Visa att ekvationen { u (x) + u (x) + λ u(x) = f(x), x [, 1] u() = u(1) =, u C 2 ([, 1]) har en entydigt bestämd lösning för λ litet. 3. Betrakta avbildningen (x 1, x 2, x 3,...) (x 1, 1 2 (x 1 +x 2 ), 1 3 (x 1 +x 2 +x 3 ),..., 1 n (x 1 +x x n ),...). Visa att detta är en begränsad linjär operator på l 2 som ej är surjektiv. 46

47 4. Låt A : H H vara en begränsad linjär operator på ett Hilbertrum H. Definiera A och visa att den är en väldefinierad begränsad linjär operator på H och att A = A. Visa slutligen att om A n A i B(H, H) då n och om alla A n är självadjungerade så är också A självadjungerad. 5. Formulera och bevisa Lax-Milgrams sats. (5p) 6. Låt T vara en linjär begränsad operator på ett Hilbertrum H med T = 1. Antag att det finns ett x H så att Tx = x. Visa att då gäller att T x = x. Motivera väl! Lycka till!! PK 47

48 TMA41/MMA4 Functional Analysis 21/211 Peter Kumlin Mathematics Chalmers & GU Gamla tentor från 2 27 lösningsförslag till några tentor 1

49 Lösningsförslag till TMA41/MAN We should show that Tf(x) = 2π sin(x t)f(t) dt, x [, 2π] defines a linear bounded and compact operator on (a) C([, 2π]), (b) L 2 ([, 2π]). Linearity: standard calculation (omitted here) Boundedness: Note that the norms in a) and b) are different! (a): Tf(x) 2π sin(x t) f(t) dt 2π sin(x t) f dt 2π f implies that T is bounded and T 2π. (b): Tf 2 = 2π 2π sin(x t)f(t) dt 2 dx 2π 2π sin(x t) 2 dt f 2 dx (2π) 2 f 2 implies that T is bounded and T 2π. Compactness: (a): This is a consequence of Arzela-Ascoli theorem. More precisely fix a bounded sequence (f n ) n=1 in C([, 2π]). Set M = sup n f n. Then i) (Tf n ) n=1 is uniformly bounded in C([, 2π]) since sup n Tf n 2πM and ii) (Tf n ) n=1 is equicontinuous on C([, 2π]) since sin x is uniformly continuous on [ 2π, 2π] and hence for fixt ǫ > there exists a δ > such that Tf n (x) Tf n (y) M provided x y < δ. 2π sin(x t) sin(y t) dt < ǫ (b): Since every operator on a Hilbert space with finitedimensional range is compact and R(T) = Span{sin x, cosx} we obtain that T is compact. 2. Consider the BVP { Lu u (x) u(x) = λ sin(u(x 2 )), x [, 1] u() = u(1) =, u C 2 ([, 1]) ( ) Calculation of the Green s function: g(x, t) = (a 1 (t)e x + a 2 (t)e x )θ(x t) + b 1 (t)e x + b 2 (t)e x where { a1 (t)e t + a 2 (t)e t = a 1 (t)e t a 2 (t)e t = 1 2 i.e. { a1 (t) = 1 2 e t a 2 (t) = 1 2 et

50 and { b1 (t) + b 2 (t) = sinh(1 t) + eb 1 (t) + e 1 b 2 (t) = i.e. { b 1 (t) = e sinh(1 t) e 2 1 b 2 (t) = e sinh(1 t) e 2 1 Hence we have g(x, t) = sinh(x t)θ(x t) Now set T : C([, 1]) C([, 1]), 2e sinh(1 t) sinh x. e 2 1 where Tu(x) = g(x, t)( λ sin(u(t2 ))) dt. From Banach s fixed point theorem we conclude that ( ) has a unique solution if T is a contraction. For u, v C([, 1]) we have Tu(x) Tv(x) = λ λ g(x, t) dt u v. g(x, t)(sin(u(t 2 )) sin(v(t 2 ))) dt Hence T is a contraction for λ < R max 1 and the desired conclusion x 1 g(x,t) dt follows. (Calculations giving an estimate for λ omitted here) 3. For n = 1, 2, 3,... we have f n = e n+1 e n, where (e n ) n=1 is an ON-basis for the Hilbert space H. Assume that Span{f n : n = 1, 2, 3,...} is not dense in H. Set S = Span{f n : n = 1, 2, 3,...}. Then S is a closed proper subspace of H. Let x S. Here i.e. But (e n ) n=1 is an ON-basis and so = x, f n = x, e n+1 x, e n, x, e n+1 = x, e n, n = 1, 2, 3,... 1 where x = Σ n=1 x, e n e n Σ n=1 x, e n 2 < by Bessel s inequality. This implies that x, e n = for all n and x =. This gives a contradiction. Hence S = H and the conclusion follows. 4. See textbook 5. See textbook 6. Let (E,, ) be an inner product space with the Riesz property. Assume that (E,, ) is not a Hilbert space. Let denote the norm on E given by x = x, x, x E. Then (E, ) is not a Banach space. Let (Ẽ, ) 3

51 denote a completion of (E, ). Here E is dense in Ẽ. Note that satisfies the parallellogram law and that x, y = 1 4 ( x + y 2 x y 2 ) + 1 4i ( x + iy 2 x iy 2 ) for all x, y E. Then satisfies the parallellogram law by continuity and so x, y = 1 4 ( x + y 2 x y 2 ) + 1 4i ( x + iy 2 x iy 2 ) defines an inner product on Ẽ that coincides with, on E. Now (Ẽ,, ) is a Hilbert space and by Riesz representation theorem has the Riesz property. Fix x Ẽ \ E and define f(x) = x, x, x E. Then f defines a bounded linear functional on E. Note that f(x) x x for x E. By the Riesz property for E there exists a x E such that f(x) = x, x for all x E. Hence x, x x =, x E. But E is dense in Ẽ so x x =. This gives x = x which yields a contradiction. Hence (E,, ) is a Hilbert space. 4

52 Lösningsförslag till TMA41/MAN Consider the BVP { Lu u (x) + u(x) = λ cos(1 + u(x)), x [, 1] u() = u () =, u C 2 ([, 1]) ( ) Calculation of the Green s function: where g(x, t) = (a 1 (t) cosx + a 2 (t) sin x)θ(x t) + b 1 (t) cosx + b 2 (t) sin x { a1 (t) cost + a 2 (t) sin t = a 1 (t) sin t + a 2 (t) cost = 1 i.e. { a1 (t) = sin t a 2 (t) = cos t and { b1 (t) = b 2 (t) = Hence we have g(x, t) = sin(x t)θ(x t). Now set T : C([, 1]) C([, 1]), where Tu(x) = g(x, t)( λ cos(1 + u(t))) dt. From Banach s fixed point theorem we conclude that ( ) has a unique solution if T is a contraction. For u, v C([, 1]) we have Tu(x) Tv(x) = λ λ g(x, t) dt u v = λ λ (1 cos 1) u v. Hence T is a contraction for λ < 1 1 cos 1 g(x, t)(cos(1 + u(t)) cos(1 + v(t))) dt x sin(x t) dt u v and the desired conclusion follows. 2. (e n ) n=1 is an ON-basis in a Hilbert space H and T is defined by T(Σ n=1 a ne n ) = Σ 1 n=1 n + 1 a n+1e n for (a n ) n=1 l 2. Clearly T B(H, H) with T = 1. An easy calculation 2 gives T (Σ n=1 b 1 ne n ) = Σ n=2 n b n 1e n. T is a compact operator since T T M as M where T M, M = 1, 2,..., are finite dimensional operators defined by T M (Σ n=1 a ne n ) = Σ M 1 n=1 n + 1 a n+1e n. 5

53 More precisely we have T T M < 1, M = 1, 2,.... M Moreover λ is an eigenvalue for T iff there exists an (eigen)vector Σ n=1 a ne n such that T(Σ n=1 a ne n ) = λσ n=1 a ne n, i.e. λa n = 1 n + 1 a n+1, n = 1, 2,... This implies that only λ = is an eigenvalue for T (with eigenvector e 1 ). Finally µ is an eigenvalue for T iff there exists an (eigen)vector Σ n=1b n e n such that T (Σ n=1 b ne n ) = µσ n=1 b ne n. This means that b 1 =, µb n = 1 n b n 1, n = 2, 3,... Hence T has no eigenvalues. This gives σ p (T) = {} and σ p (T ) =. 3. Riesz representation theorem implies that there are uniquely defined y k H, k = 1, 2,..., n, such that f k (x) = x, y k all x H, where H is a Hilbert space. Moreover the fact that f 1, f 2,...,f n are linearly independent in B(H, ) implies that y 1, y 2,...,y n are linearly independent 1 in H (easy to show). Now for each l {1, 2,..., n} consider the set Y l = {y k : k l}. We see that f l Yl since otherwise f l (x) = for all x Y l. This would imply that {y k : k l} {y l } and hence Span{y l } Span{y k : k l}, which contradicts the linearly independence of y 1, y 2,...,y n Finally, for each l {1, 2,..., n} pick an x l Y l such that f l (x l ) = 1. These x l :s will satisfy the properties stated in the problem. 4. See textbook 5. See textbook 6. Let H be a Hilbert space and let T : H H be a linear mapping with the following property: We should prove that T is bounded 2. x n x in H Tx n Tx in H. Assume that T is not bounded. Then there exists a sequence (x n ) n=1 such that x n in H but Tx n T = in H. Without loss of generality (easy to show) we may assume that 1 y 1, y 2,..., y n does not need to be pairwise orthogonal. 2 This can be done using the closed graph theorem, see the lecture notes on spectral theory, but I have not discussed that theorem in class. 6

54 (a) Tx n = 1 for n = 1, 2,..., (b) x n 2 n for n = 1, 2,..., (c) Tx n in H. Set y n = Tx n for n = 1, 2,... and choose an increasing sequence (n l ) l=1 of integers as follows: Set n 1 = 1. For l = 2, 3,... let n l have the property Σ l 1 k=1 y n k, y m 1 4 all m n l. The existence of (n l ) l=1 follows from the fact that y n. This implies that Σ M l=1 y n l 2 = Σ M l=1 y n l, y nl + Σ M k,l=1,k l y n k, y nl M 2Σ 1 k<l M y nk, y nl = Σ M l=1 (1 2Σl 1 k=1 y n k, y nl ) M 2. Now y nk = Tx nk and x nk 2 k. Hence Σ M k=1 x n k x for some x H but T(Σ M k=1 x n k ) = Σ M k=1 Tx n k T x since 3 Σ M k=1 Tx n k 2 M as M. 2 Contradiction! Hence T is bounded. 3 Every weakly convergent sequence is bounded 7

55 Lösningsförslag till TMA41/MAN Consider the BVP { u (x) = 1 (1 1 ), x [, 1] 5 1+u 4 (x) u() = u(1) =, u 2 ([a, 1]) ( ) Step 1: Calculate the Green s function g(x, t) for Lu = u, u() = u(1) =. Here g(x, t) = (a 1 (t) 1 + a 2 (t)x)θ(x t) + b 1 (t) 1 + b 2 (t)x where { a1 (t) + a 2 (t)t = a 2 (t) = 1 i.e. a 1 (t) = t a 2 (t) = 1 and { g(, t) =, < t < 1 g(1, t) =, < t < 1 which yields Step 2: Set Tu(x) = g(x, t) = i.e. b 1 (t) = b 2 (t) = t 1 { (t 1)x x < t 1 (x 1)t t < x 1 g(x, t)( )dt, u C([, 1]). 1 + u 4 (t) If T : C([, 1]) C([, 1]) is a contraction, then ( ) has a unique solution. T is a contraction since for u, v C([, 1]), Tu(x) Tv(x) 1 5 {mean value thm } u v. 1 g(x, t) 1 + u 4 (t) v 4 (t) dt g(x, t) dt u v Hence Tu Tv 4 1 u v and so T is a contraction. The Banach fixed point theorem yields the result. 2. (e n ) n=1 ON-basis on H implies (f k ) k= is an ON-basis on H where f = e 1, f k = e 2k+1, k = 1, 2,..., f k = e 2k, k = 1, 2,..., so every x H has a unique representation k= a kf k, where k= a k 2 < (more precisely a k = x, f k, k ). Clearly, S is well-defined and S( a k f k ) 2 = k= k= a k f k 2 = k= a k 2 k 8 a k f k+1 2 = k= a k 2

56 by Parseval s formula. Hence S is an isometry and S = 1. S has no eigenvalues since if λ = then a k = all k λ then a k = λa k+1 all k and hence unless k= a k 2 =. k= a k 2 = 3. (e k ) k=1 is a sequence in H, where e k = 1 for all k. We want to show that x, e k 2 x 2 (1 + ( e k, e l 2 ) 1/2 ). k k l Note that x, e k 2 = x, e k x, e k = x, Σ k x, e k e k k k x Σ k x, e k e k. Moreover k x, e k e k 2 = k,l x, e k x, e l e k, e l k,l = k x, e k x, e l e k, e l = x, e k 2 + k l( x, e k x, e l ) e k, e l k x, e k 2 + ( k l( x, e k 2 x, e l 2 ) 1/2 ( k l e k, e l 2 ) 1/2 k x, e k 2 + ( k,l x, e k 2 x, e l 2 ) 1/2 ( k l e k, e l 2 ) 1/2 = = k x, e k 2 (1 + ( k l e k, e l 2 ) 1/2 ). This yields the result and 6. Theory from book (and lectures). 9

57 Lösningsförslag till TMA41/MAN Consider the BVP { Lu = u (x) u(x) = 1 2 (1 + u(x2 )), x [, 1] u() = u () =, u C 2 ([, 1]) ( ) Calculation of Green s function g(x, t) = (a 1 (t)e x + a 2 (t)e x )θ(x t) + b 1 (t)e x + b 2 (t)e x where { a1 (t)e t + a 2 (t)e t = a 1 (t)e t a 2 (t)e t = 1 i.e. a 1 (t) = 1 2 e t a 2 (t) = 1 2 et and { b1 (t) + b 2 (t) = b 1 (t) b 2 (t) = i.e. Hence we have g(x, t) = sinh(x t)θ(x t). Now set T : C([, 1]) C([, 1]), b 1 (t) = b 2 (t) = where Tu(x) = g(x, t)( 1 2 (1+u(t2 )))dt. From Banach s fixed point theorem we conclude that ( ) has a unique solution if T is a contraction. For u, v C([, 1]) we have Tu(x) Tv(x) = 1 2 g(x, t) 1 2 (u(t2 ) v(t 2 ))dt g(x, t) dt u v = 1 2 (cosh k 1) u v 1 4 (e + 1 e 2) u v. }{{} <1 Here T is a contraction and the statement follows. 2. Set, for f L 2 ([a, b]), Tf(x) = 1 b a Tf L 2 ([a, b]) since b f 2 L = 1 ( 2 b a 1 = ( b a )2 1 ( b a )2 a b a b T linear: easy to show. a b a b a (b a) b f(x)dx, x [a, b]. a f(x) dx ) 2 dt = f(x)dx 2 dt {Hölder} b a f(x) 2 dxdt = b T bounded: see above. In particular we get T 1. a f(x) 2 dt = f 2 L 2. To show that T is an orthogonal projection it suffices to show that T 2 = T and T = T. 1

58 (a) Take f L 2 ([a, b]). Then b (T 2 1 f)(x) = T( b a a = 1 b f(t)dt = (Tf)(x), b a Hence T 2 = T. (b) Take f, g L 2 ([a, b]). We obtain Hence T = T. a b 1 Tf, g = a b a b 1 = f(x) b a = The statement is proved. a b a f(x) f(t)dt) = 1 b b a a 1 b a 1 b a all x [a, b]. b a b a b 3. Let h C([, 1] [, 1]) be real-valued and a b a f(x)dx g(t)dt = g(t)dtdx = g(t)dtdx = f, Tg f(x)dtds = h(x, y) = h(y, x) > all x, y [, 1]. ( ) Set Tf(x) = h(x, y)f(y)dy, x [, 1], for f L2 ([, 1]). We want to show that T has an eigenvalue λ = T whitch is simple. (All eigenvalues λ satisfy λ T ). Since the kernel is continuous and satisfies ( ) we see that T is a compact, self-adjoint operator or L 2 ([, 1]) and hence has an eigenvalue λ Ê with λ = T. Since h > we see that λ = T (see first and second observation below). It remains to prove that this eigenvalue is simple. First observation: f eigenfunction for λ f C([, 1]) which follows from Lebesgues dominated convergence. Then Second observation: f eigenfunction for λ f has constant sign, say f, since if f changes sign, then λ f = T f = Tf < T f T f = T f. Moreover we can conclude that f > since h >. Third observation: f 1, f 2 eigenfunction for λ f 1 = αf 2 for some α. To see this assume that it is false and set s(α) = {x [, 1] : f 1 (x) αf 2 (x) }, α, where E denotes the measure of the set E. Here s() = 1, s(α) decreasing and lim a + s(α) =. Also s(α) = 1 for α [, α] for some α > since 11

59 f 1, f 2 C([, 1]) and f 1, f 2 > on [, 1]. Then there exists < α < α 1 such that 1 > s(α ) s(α 1 ) >. ) This means that f 1 (x) < α f 1 (x) f 1 (x) α f 2 (x) on a set of positive measure on a set of positive measure but since also f 1 (x) α 1 f 2 (x) on a set of positive measure, together with < α < α 1 and f 2 > we see that f 1 (x) > α f 2 (x) on a set of positive measure. Then f = f 1 α f 2 is an eigenfunction for λ (note that f ) that changes sign. Contradiciton! The statement follows. ) It cannot happen that β > sth. { s(α) = 1 for α < β s(α) = for α > β easy to see. and if there exists a β > sth then f 1 = βf 2. s(α) = 1 for s(α) = for α β α > β 12

60 Lösningsskisser till tentamen i TMA41/MAN67, We will prove that { u (x) + u(x) = u(x), x π 2+u 2 (x) 2 u() = u( π) =, u 2 C2 ([, π]) 2 has a unique solution. { u 1. Determine the Green s function for + u = F u() = m( π) = : 2 Set e(x, t) = a 1 (t) cosx + a 2 (t) sin x, where e(t, t) = and e x(t, t) = 1. This gives e(x, t) = sin(x t). The Green s function takes the form g(x, t) = sin(x t)θ(x t) + b 1 (t) cosx + b 2 (t) sin x. Here g(, t) = g( π 2, t) = for < t < π 2 implies that g(x, t) = sin(x t)θ(x t) sin x cos t = { cosxsin y, x > t = sin x cost, x < t We see that g(x, t) for all x, t [, π 2 ]. 2. Set { (Tu)(x) = π 2 u C([, π]) 2 g(x, t) u(t) 2+u 2 (t) dt, x π 2 The boundary value problem has a unique solution iff T has a unique fixed point. For u, v C([, π ]) we get 2 π 2 u(t) (Tu)(x) (Tv)(x) g(, t) 2 + u 2 (t) v(t) 2 + v 2 (t) dt {mean value theorem} 1 2 π 2 g(x, t) dt u v π 4 u v This shows that T is a contraction on the space C([, π ]) and the conclusion follows from Banach s fixed point 2 theorem. 2. Set Tf(x) = (x + y)f(y)dy for f L2 ([, 1]). T is bounded since Tf 2 L = 2 (x + y)f(y)dy 2 dx {Hölder} (x + y) 2 dy f 2 L 2dx = 7 6 f 2 L 2 13

61 7 and hence T. 6 To calculate T we observet hat T is a compact, self-adjoint operator on the Hilbert space L 2 ([, 1]) and hence T = sup λ eigenvalue to T Note that Tf(x) = f(y)dy x + yf(y)dy is a polynomial of degree 1 and hence λ is an eigenvalue to T with eigenfunction ax + b if λ(ax + b) = (ay + b)dy x + λ. y(ay + b)dy, x [.1]. i.e., { λa = 1 2 a + b λb = 1 3 a b This system has a non-trivial solution iff 1 2 λ λ =. 3 2 We obtain the eigenvalues λ 1,2 = 1 2 ± 1 3 and so T = From problem 1 we can restate the problem as to show that the mapping, π 2 Tu(x) = λ u(t) g(x, t) 2 + u 2 (t) dt, x π 2, for u C([, π]), has a fixed point in C([, π ]) for all λ Ê. For λ large 2 2 enough we cannot refer to the Banach s fixed point theorem since T is no longer a contraction. Instead we can use the Schander s fixed point theorem. Fix a λ Ê. Note that g(x, t) is a continuous function for x, t π 2 is a bounded function for u Ê (note that λ is fixed.) and f(u) λ u 2+u 2 We will choose a closed convex set S C([, π ]) such that the mapping 2 T : S S is continuous and the image set T(S) is relatively compact in C([, π]). 2 Take S = {u C([, π ]) : u D}, where D is to be chosen such that 2 T(S) S. Since g is continuous on the compact set {(x, y) : x, t π } it 2 is bounded on this set and so sup g(x, t)f(u) = E < x,t π 2 u Ê which yields (Tu)(x) π E for all x [, π ] and u C([, π ]). Hence T(S) S if D = πe. 2 14

62 Now it remains to prove that T(S) is relatively compact in C([, π ]) and 2 T S S is continuous. The first statement follows from the Arzela-Aschi Theorem, since T(S) is uniformly bounded and equicontinuous (the latter follows from the uniform continuity of g on {(x, t) : x, t π }), and the 2 second statement follows from the uniform continuity of f(y) for D u D and the boundedness of g. The existence of a fixed point for T now follows from Schander s fixed point theorem. (Remark: As a matter of fact one trivially observes that u = is a solution to the problem. However to treat the problem with the RHS in the differential equation replaced by the original one +1 is harder but the method above yields a solution.) 4. See textbook. 5. T : X X, where X is a real normal space, is continuous and satisfies T(x + y) = T(x) + T(y) for all x, y X. ( ) We shall prove that T(λx) = λt(x) for all x X and λ Ê ( ) It follows from ( ) that T() = since T() = T( + ) = 2T(). T(nx) = nt(x) for positive integrs n since T(nx) = T(x + (n 1)x) = T(x) + T((n 1)x) =... = nt(x) T( 1 x) = 1 T(x) for positive integers n since n n T(x) = T(n( 1 n x)) = nt(1 n x). Combining these observations we see that T(λx) = λt(x) for all x X and λ É. To prove ( ) we fix a λ Ê and an x X and take a sequence λ n É, n = 1, 2,..., such that λ n λ in Ê. This implies that λ n x λx in X. It also implies λ n T(x) λt(x) in X. Since T is continuous we conclude that T(λ n x) T(λx) in X. But, T(λ n x) = λ n T(x) λt(x) in X and so The statement is proved. T(λx) = λt(x) 6. Let (x n ) n=1 be an ON-basis in H and (y n) n=1 an ON-sequence in H such that n=1 x n y n 2 < 1. We shall show that also (y n ) n=1 is an ON-basis. Set S = span{y n : n = 1, 2,...}. Then S is a closed subspace of H. It remains to prove that S = H. 15