- For f,g ∈ Clet= ∫
_{0}^{1}fgdx. Is this an inner product space? Is it a Hilbert space? What does the Cauchy Schwarz inequality say in this context? - Suppose the following conditions hold.
(24.53) (24.54) For a,b ∈ ℂ and x,y,z ∈ X,

(24.55) These are the same conditions for an inner product except it is no longer required that

= 0 if and only if x = 0. Does the Cauchy Schwarz inequality hold in the following form? - Let S denote the unit sphere in a Banach space, X,
Show that if Y is a Banach space, then A ∈ℒ

is compact if and only if Ais precompact, Ais compact. A ∈ℒis said to be compact if whenever B is a bounded subset of X, it follows Ais a compact subset of Y. In words, A takes bounded sets to precompact sets. - ↑ Show that A ∈ℒis compact if and only if A
^{∗}is compact. Hint: Use the result of 3 and the Ascoli Arzela theorem to argue that for S^{∗}the unit ball in X^{′}, there is a subsequence,⊆ S^{∗}such that y_{n}^{∗}converges uniformly on the compact set, A. Thusis a Cauchy sequence in X^{′}. To get the other implication, apply the result just obtained for the operators A^{∗}and A^{∗∗}. Then use results about the embedding of a Banach space into its double dual space. - Prove the parallelogram identity,
Next suppose (X,||||) is a real normed linear space and the parallelogram identity holds. Can it be concluded there exists an inner product (⋅,⋅) such that ||x|| = (x,x)

^{1∕2}? - Let K be a closed, bounded and convex set in ℝ
^{n}and let f : K → ℝ^{n}be continuous and let y ∈ ℝ^{n}. Show using the Brouwer fixed point theorem there exists a point x ∈ K such that P= x. Next show that≤ 0 for all z ∈ K. The existence of this x is known as Browder’s lemma and it has great significance in the study of certain types of nolinear operators. Now suppose f : ℝ^{n}→ ℝ^{n}is continuous and satisfiesShow using Browder’s lemma that f is onto.

- Show that every inner product space is uniformly convex. This means that if x
_{n},y_{n}are vectors whose norms are no larger than 1 and if→ 2, then→ 0. - Let H be separable and let S be an orthonormal set. Show S is countable. Hint: How far apart are two elements of the orthonormal set?
- Suppose {x
_{1},,x_{m}} is a linearly independent set of vectors in a normed linear space. Show spanis a closed subspace. Also show every orthonormal set of vectors is linearly independent. - Show every Hilbert space, separable or not, has a maximal orthonormal set of vectors.
- ↑ Prove Bessel’s inequality, which says that if {x
_{n}}_{n=1}^{∞}is an orthonormal set in H, then for all x ∈ H,||x||^{2}≥∑_{k=1}^{∞}|(x,x_{k})|^{2}. Hint: Show that if M = span(x_{1},,x_{n}), then Px = ∑_{k=1}^{n}x_{k}(x,x_{k}). Then observe ||x||^{2}= ||x − Px||^{2}+ ||Px||^{2}. - ↑ Show S is a maximal orthonormal set if and only if spanis dense in H, where spanis defined as
- ↑ Suppose {x
_{n}}_{n=1}^{∞}is a maximal orthonormal set. Show thatand ||x||

^{2}= ∑_{i=1}^{∞}|(x,x_{i})|^{2}. Also show (x,y) = ∑_{n=1}^{∞}(x,x_{n})(y,x_{n}). Hint: For the last part of this, you might proceed as follows. Show thatis a well defined inner product on the Hilbert space which delivers the same norm as the original inner product. Then you could verify that there exists a formula for the inner product in terms of the norm and conclude the two inner products,

andmust coincide. - Suppose X is an infinite dimensional Banach space and suppose
are linearly independent with

= 1. By Problem 9 span≡ X_{n}is a closed linear subspace of X. Now let zX_{n}and pick y ∈ X_{n}such that≤ 2distand letShow the sequence

satisfies≥ 1∕2 whenever k < n. Now show the unit ballin a normed linear space is compact if and only if X is finite dimensional. Hint: - Show that if A is a self adjoint operator on a Hilbert space and Ay = λy for λ a
complex number and y≠0, then λ must be real. Also verify that if A is self
adjoint and Ax = μx while Ay = λy, then if μ≠λ, it must be the case that
= 0.
- Theorem 24.9.8 gives the the existence and uniqueness for an evolution equation of the
form
where g is in C

^{1}for H a Banach space. Recall Λ was the generator of a continuous semigroup, S. Generalize this to an equation of the formwhere L is a continuous linear map. Hint: You might consider Λ + L and show it generates a continuous semigroup. Then apply the theorem.

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