Question

Show that every separable Banach space has an equivalent URED norm.

Short answer

By the Lindenstrauss–Tzafriri theorem, any separable Banach space can be given an equivalent uniformly convex (and hence URED) norm.

Step-by-step solution

Understand Key Definitions

In this problem, a Banach space is a complete normed vector space. A space is separable if it contains a countable dense subset. A norm is URED (Uniformly Rotund in Every Direction) if, loosely speaking, the norm is strictly convex in every direction.

Use the Lindenstrauss–Tzafriri Theorem

According to the Lindenstrauss–Tzafriri theorem, every separable Banach space can be renormed to be uniformly convex. Note that a uniformly convex norm is also URED.

Renorm the Separable Banach Space

Given a separable Banach space, by the Lindenstrauss–Tzafriri theorem, there exists an equivalent norm on this space which makes it uniformly convex. Because uniformly convex norms are URED, the Banach space thus has an equivalent URED norm.

Key concepts

URED Norms

In functional analysis, the concept of URED (Uniformly Rotund in Every Direction) norms is essential. A norm is considered URED if it maintains strict convexity in every direction. This means that any two distinct points in the space will have a midpoint that lies 'inside' the unit ball, except at the boundary points. This property ensures that the norm doesn't have flat spots and gives more structure to the geometry of the space. The URED norm is an excellent tool to study the geometry of Banach spaces, particularly because it helps in preventing the construction of complex substructures within the space.

Lindenstrauss–Tzafriri Theorem

The Lindenstrauss–Tzafriri theorem is a cornerstone in the study of Banach spaces. It states that every separable Banach space can be renormed to become uniformly convex. This means we can change the norm on the space such that the new norm is uniformly convex. Uniform convexity implies that for any two points, the midpoint between them lies strictly inside the unit ball, except if the points are identical. A significant outcome of this theorem is its application to URED norms. Since uniformly convex norms are also URED, this theorem guarantees that separable Banach spaces can always adopt an equivalent URED norm, simplifying analyses that involve these spaces.

Uniformly Convex Spaces

Uniformly convex spaces are Banach spaces equipped with a norm that ensures extra geometric properties. If a space is uniformly convex, it means that any sequence of vectors converging to the same vector will have their midpoints uniformly bounded away from the edge of the unit ball. To formally describe this, a space is uniformly convex if for any \( \varepsilon > 0\), there exists a \( \delta > 0\) such that if \( \| x \| = 1 \) and \( \| y \| = 1 \), then \( \|x+y \)/2\| < 1 - \( \delta\), whenever \( \|x-y\| \) is greater than \( \varepsilon\).

• This property is essential for the study of reflexivity and optimization in Banach spaces.
• It ensures stronger convergence properties which can be useful in numerical analysis and other applications.

Thus, the ability to renorm separable Banach spaces to be uniformly convex simplifies many theoretical aspects and computations in functional analysis.