Characteristic Inequalities in Banach Spaces and Applications
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Characteristic Inequalities in Banach Spaces and Applications

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– Characteristic Inequalities in Banach Spaces and Applications –

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Abstract

Basic notions of functional analysis: In this chapter, we recall some definitions and results from linear functional analysis. Proposition 1.1.1 (The Parallelogram Law) Let X be an inner product space.

Then for arbitrary x, y ∈ X, kx + yk 2 + kx − yk 2 = 2 kxk 2 + kyk 2  . Theorem 1.1.1 (The Riesz Representation Theorem) Let H be a Hilbert space and let f be a bounded linear functional on H.

Then there exists a unique vector y0 ∈ H such that f(x) = hx, y0i for each x ∈ H and ky0k = kfk.

Theorem 1.1.2 Let X be a reflexive and strictly convex Banach space, K be a nonempty, closed, and convex subset of X. Then for any fixed x ∈ X there exists a unique m∗ ∈ K such that kx − m∗ k = inf k∈K kx − kk. Proof.

Introduction

This completes the proof. We now show that m∗ ∈ K is unique. Indeed, if x ∈ K then m∗ = x and hence it is unique.

Suppose x ∈ Kc and m 6= n such that kx − mk = kx − nk ≤ kx − kk ∀k ∈ K, then 1 kx − mk k 1 2 ((x − m) + (x − n))k < 1. This implies that kx − 1 2 (m + n)k < kx − mk and this contradict the fact that m is a minimizing vector in K.

Therefore m∗ ∈ K is unique. Corollary 1.1.1 Let X be a uniformly convex Banach space and K be any nonempty, closed and convex subset of X. Then for arbitrary x ∈ X there exists a unique k ∗ ∈ K such that kx − k ∗ k = inf k∈K kx − kk.

Remark If H is a real Hilbert space and M is any nonempty, closed, and convex subset of H then in view of the above corollary, then there exists a unique map PM : H → M defined by x 7→ PMx, where kx − PMxk = inf m∈M kx − mk. This map is called the projection map.

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