Question

Show that a norm \(\|\cdot\|\) of a Banach space \(X\) is uniformly smooth if and only if for every \(\varepsilon>0\) there is \(\delta>0\) such that \(\|x+y\|+\|x-y\| \leq 2+\varepsilon\|y\|\) whenever \(\|x\|=1,\|y\|<\delta\)

Short answer

The norm \( \| \cdot \| \) in a Banach space is uniformly smooth if and only if for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( \| x + y \| + \| x - y \| \leq 2 + \varepsilon \| y \| \) whenever \( \| x \| = 1 \) and \( \| y \| < \delta \).

Step-by-step solution

Define the terms

To show that the norm \( \| \cdot \| \) of a Banach space \(X\) is uniformly smooth if and only if a certain condition is met, first understand the given terms. Uniform smoothness refers to the property where for every \( \varepsilon > 0\), there is some \( \delta > 0\) such that a specific inequality holds.

State the inequality condition

The condition that needs to be satisfied is \( \|x + y\| + \|x - y\| \leq 2 + \varepsilon \|y\|\) whenever \( \|x\| = 1 \) and \(\| y \| < \delta \).

Implication for uniform smoothness

For uniform smoothness of the norm, this inequality suggests that the sum of norms of \( x + y \) and \( x - y \) is bounded by \( 2 + \varepsilon \| y \| \).

Proof Sketch

To prove the sufficiency of this condition, assume uniform smoothness and show that it implies the given inequality for small \( \delta \). Conversely, assume that the inequality holds and show that this implies uniform smoothness.

Show sufficiency

Suppose \( \| \cdot \| \) is uniformly smooth. By definition, for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( \| x + y \| + \| x - y \| \leq 2 + \varepsilon \| y \| \).

Show necessity

Conversely, suppose the condition holds. This implies uniform smoothness because it ensures that the difference norms are controlled and smooth transitions occur for small perturbations. This satisfies the criteria for uniform smoothness.

Key concepts

Banach Space

A Banach space is a complete normed vector space. This means that every Cauchy sequence (a sequence where the elements get arbitrarily close to each other) in the space has a limit that also belongs to the space. Banach spaces are named after the Polish mathematician Stefan Banach who developed the theory in the early 20th century.

In more practical terms:

• A vector space has a norm, which is a function that assigns a length to each vector.
• Completeness ensures that sequences of vectors that should converge will indeed converge to a vector inside the space.

Banach spaces are foundational in functional analysis and are used extensively in solving differential and integral equations.

Uniform Smoothness

Uniform smoothness of a norm in a Banach space ensures a certain level of 'smoothness' in how the norm behaves. Formally, a norm \( \cdot \| \) is uniformly smooth if, for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that \( \|x + y \| + \|x - y \| \leq 2 + \varepsilon \|y \| \) whenever \( \|x \| = 1\) and \( \| y \| < \delta \).

This definition implies two things:

• The norm is 'smooth' in the sense that small changes in a vector result in predictably small changes in its norm.
• The degree of this smoothness can be controlled by adjusting \( \varepsilon \) and \( \delta \).

If a norm is uniformly smooth, it generally means that the Banach space can handle small perturbations well, which is crucial in applications requiring stability and sensitivity analysis.

Norm Inequalities

Norm inequalities play an essential role in functional analysis and Banach spaces. They help bound and compare the sizes of vectors. For example, in our given problem, we examine the inequality \( \|x + y \| + \|x - y \| \leq 2 + \varepsilon \|y \| \).

Let's break it down

• The left-hand side \( \|x + y \| + \|x - y \| \) represents a combination of norms of vector sums and differences.
• The right-hand side \( 2 + \varepsilon \|y \| \) shows a linear relationship plus a small perturbation term.

Understanding these inequalities is crucial for grasping how norms behave in different scenarios. They provide a mathematical way to express smoothness, stability, and continuity, which are key properties in many areas of analysis and applied mathematics.

Functional Analysis

Functional analysis is a branch of mathematics that studies spaces of functions, and the operators acting on them. Banach spaces are central structures within this field, providing the natural setting for many problems involving infinite-dimensional vector spaces.

Here's why functional analysis is important

• Optimization: Many optimization algorithms in machine learning and economics rely on principles from functional analysis.
• Differential Equations: Solutions to differential equations often reside in Banach spaces, making the analysis of these spaces crucial.
• Quantum Mechanics: The mathematics of quantum mechanics is deeply rooted in the principles of functional analysis.

Functional analysis helps mathematicians and scientists understand and solve complex problems involving functions, transformations, and the spaces they inhabit. Concepts like uniform smoothness provide the tools needed to study and manipulate these functions in a rigorous way.