Unit for Applied Numerical Analysis - M15 Department of Mathematics Technische Universität München

Dr. Frank Filbir Winter Semester 2016/2017 Dr. Sara Krause-Solberg

Functional Analysis

Exercise Sheet 10 - Frohe Weihnachten!

Exercise 10.1 (Baire categories): Let (V, P), (W, Q) be Fréchet spaces over F. Suppose T : V → W is a closed linear mapping. Prove that the following statements are equivalent:

(i) rg(T ) is of category II in W ,

(ii) T is surjective.

Exercise 10.2 (): 1 1 ˆ 1 R π −i k t For f ∈ L ([−π, π], 2π dt) define ck(f) = f(k) = 2π −π f(t) e dt, k ∈ Z. Suppose there is a 1 1 k · k on L ([−π, π], 2π dt) such that

1 1 (i) (L ([−π, π], 2π dt), k · k) is a ,

1 1 (ii) ck is a continuous linear functional on (L ([−π, π], 2π dt), k · k) for every k ∈ Z.

1 1 Prove that k · k and k · k1 are equivalent norms on (L ([−π, π], 2π dt).

Exercise 10.3 (Multiplication operators): p a) Let 1 ≤ p, q < ∞ and let a = (αn)n∈N be a sequence s.t. for all x = (ξn)n∈N ∈ ` we have q p q y = (αn ξn)n∈N ∈ ` . Prove that the multiplication Ma : ` → ` ,Max = y is linear and continuous.

p b) Let 1 ≤ q < p < ∞ and let a = (αn)n∈N be a sequence s.t. for all x = (ξn)n∈N ∈ ` we have q p q y = (αn ξn)n∈N ∈ ` . Define Ma as in a). Prove that Ma : ` → ` is linear and continuous, r a ∈ ` and kMak = kakr, where 1/p + 1/r = 1/q.

(r−q)/q [Hint: a) Use the closed graph theorem. b) Let ξn ∈ F be so that |ξn| = |αn| . Consider the action of Ma Pk r 1/r on xk = (ξ1, ξ2, . . . , ξk, 0, 0,... ) and estimate the norm kMaxkkq to get ( j=1 |αj | ) ≤ kMak. Show the upper estimate by using Hölder’s inequality.]

Exercise 10.4 (Lebesgue space): Let Lp = Lp([0, 1], dt), where dt denotes the Lebesgue measure on the unit interval [0, 1]. Suppose E ⊂ Lp, 1 ≤ p < ∞, is a closed linear subspace with E ⊂ L∞. Prove the following statements:

(i) The inclusion operator ι : E → L∞ is continuous.

(ii) There is a constant M > 0 such that kfk∞ ≤ M kfk2 for f ∈ E.

1 Hint: For (ii) use (i) and the fact that kfkp = R |f(t)|2 |f(t)|p−2 dt for p ∈ ]2, ∞[. [ p 0 ]

Page 1 of 3 Exercise 10.5 (Closed subspace of continuous functions): Let E ⊂ C([0, 1]) be a closed consisting only of C1([0, 1|) functions. Prove that E is finite-dimensional.

0 0 [Hint: Apply the closed graph theorem to show that D : E → C([0, 1]), Df = f is bounded and obtain kf k∞ ≤ M for all f ∈ B¯E := {f ∈ E : kfk ≤ 1}. Make use of the Arzela-Ascoli Theorem to show that B¯E is compact. Use the fact that the closed unit norm ball B¯(0, 1) = {x ∈ V : kxk ≤ 1} in a normed (V, k · k) is compact if and only if V has finite .]

Exercise 10.6 (Schauder ):

Let (V, k · k) be a Banach space over F. A sequence (xk)k∈ ⊂ V is called of V if Ä N ä for every x ∈ V there is a uniquely determined sequence ξk(x) ⊂ F such that k∈N ∞ X x = ξk(x) xk. k=1

a) Assume (xk)k∈N is a Schauder basis of V and define a sequence of linear operators by n X Pn : V → V,Pnx = ξk(x) xk, n ∈ N. k=1

(i) Check that Pn is a linear projection operator onto span{xk : 1 ≤ k ≤ n}.

(ii) For x ∈ V let |||x||| = sup kPnxk. n Show that |||·||| is a norm on V which is equivalent to k · k and that (V, |||·|||) is a Banach space.

(iii) Prove that Pn is continuous and uniformly bounded, i.e. supn kPnk < ∞.

(iv) Prove that the coefficient functionals ξk : V → F, x 7→ ξk(x) are linear, continuous, and fulfil the biorthogonality relation ξk(xj) = δk,j..

b) Prove that a sequence of vectors (xn)n∈N ⊂ V is a Schauder basis iff the following conditions hold.

(i) xn 6= 0 for all n ∈ N.

(iii) There is a constant K > 0 so that for every choice (ξn)n∈N ⊂ F and all n, m ∈ N, n < m, n m X X ξ x ≤ K ξ x . k k k k k=1 k=1

(iii) c`k·kspan{xn : n ∈ N} = V .

k c) Define a sequence of functions (φn)n∈N for k ∈ N, l = 1, 2,..., 2 by  1, t ∈ [(2l − 2) 2−k−1, (2l − 1) 2−k−1],   −k−1 −k−1 φ2k+l(t) := −1, t ∈ ](2l − 1)2 , 2l 2 ],   0, otherwise.

Page 2 of 3 p Prove by using the result in b) that (φn)n is a Schauder basis for all L ([0, 1]), 1 ≤ p < ∞.

[Hint: Consider the dyadic intervals of the form [l 2−k, (l + 1) 2−k] to show b)(iii). To show b)(ii) consider Pn Pn+1 the functions f(t) = k=1 αk φk(t) and g(t) = k=1 αk φk(t) and analyze the difference in order to obtain kfkp ≤ kgkp.]

Remark. 1) Note that a Schauder basis (xn)n is a sequence, i.e. it comes with a special ordering. It is in general not true that if (xn)n is a Schauder basis of V then for any permutation σ : N → N the system (xσ(n))n∈N is also a Schauder basis. If this is the case the Schauder basis is called unconditional.

2) Don’t confuse Schauder basis with Hamel basis (also called algebraic basis). A Hamel basis {eλ : λ ∈ Λ} for V is a set of linearly independent vectors with span{eλ : λ ∈ Λ} = V . This means every x ∈ V is uniquely representable by a finite of elements in {eλ : λ ∈ Λ}. It can be shown using Baire’s Theorem that a Hamel basis for an infinite-dimensional Banach space must be uncountable. Try to do it.

3) The system in c) is called Haar system. Integrating the elements of the Haar system leads to the system

Z t ϕ1(t) = 1, ϕn(t) = φn−1(s) ds, 0 which is the famous Schauder system. It is a Schauder basis for the space C([0, 1]) (but not a unconditional Schauder basis). Draw some graphs and maybe try to prove the Schauder basis property. Clearly, every Banach space which has a Schauder basis is separable. So we see from the constructions above that all spaces Lp([0, 1]), 1 ≤ p < ∞ and the space C([0, 1]) are separable Banach spaces. However, the space L∞([0, 1]) is not separable, consequently there does not exist a Schauder basis for this space.

4) There are several questions poping up.

a) Does every separable Banach space has a Schauder basis?

b) Does every infinite dimensional Banach space contain a basic sequence, i.e. a sequence which is a Schauder basis for the closure of its linear span?

c) Does every separable Banach space embed isometrically into a Banach space with a Schauder basis?

Problem a) was open for about 40 years. This problem was finally answered negatively by Per Enflo in 1972, published in 1973 (A counterexample to the approximation problem in Banach spaces. Acta Mathematica 130 (1)). The prize awarded to Per Enflo for this achievement was a live goose. You can find a bit more facts about the history on Wikipedia (search for Per Enflo). The question b) was answered positively by Stanislaw Mazur in 1933. Question c) was answered positively by and Stanislaw Mazur. They showed that every separable Banach space can be isometrically embedded into C([0, 1]) (Zur Theorie der linearen Dimensionen. Studia Mathematica 4, 100-112, 1933).

This sheet will be discussed from Monday, January 9 on.

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