Question

A norm \(\|\cdot\|\) of a Banach space \(X\) is called uniformly rotund in every direction \((U R E D)\) if for every \(z \in S_{X}\) and all bounded sequences \(\left\\{x_{n}\right\\},\left\\{x_{n}\right\\} \subset X\) such that \(2\left\|x_{n}\right\|^{2}+2\left\|y_{n}\right\|^{2}-\left\|x_{n}+y_{n}\right\|^{2} \rightarrow 0\) and \(x_{n}-y_{n}=\lambda_{n} z\) for some \(\lambda_{n}\), we have \(\lambda_{n} \rightarrow 0\) Let \(\Gamma\) be an uncountable set. Show that \(c_{0}(\Gamma)\) has no equivalent URED norm.

Short answer

$c_0(\textbackslashGamma) does not support an equivalent URED norm under the given conditions.

Step-by-step solution

- Understand Uniformly Rotund in Every Direction (URED) Norm

A norm is called uniformly rotund in every direction if for every element in the unit sphere and all bounded sequences meeting certain conditions, a specific convergence is observed.

- Recall Definition of the Unit Sphere

The unit sphere in a Banach space \(X\), denoted as \(S_X\), is the set of all elements \(x\) in \(X\) such that \(orm{x} = 1\).

- Identify Conditions for URED Norm

Given sequences \(orm{x_n} = orm{y_n} = 1\) and the condition: \(2 orm{x_n}^2 + 2 orm{y_n}^2 - orm{x_n + y_n}^2 \rightarrow 0\) and \(x_n - y_n = \theta_n z\) where \(\theta_n \rightarrow 0\).

- Analyze the Structure of \(c_0(\boldsymbol{\boldsymbol{\textdollar}\textbackslashtextbackslash textdollar})\) space

\(c_0(\boldsymbol{\textdollar}\textbackslashtextbackslashGamma \textdollar)\) is the space of all functions \(x\) from \(\boldsymbol{\textdollar}\textbackslashtextbackslashGamma \textdollar\) to \(\boldsymbol{\textdollar}\textbackslashtextbackslashmathbb{R} \textdollar\) such that \(\textbackslashtextbackslashlim\textunderscore\textbackslashtextbackslash gamma \rightarrow \textbackslashtextbackslashinfty \textbackslashtextbackslash x(gamma) = 0\).

- Show \(c_0(\boldsymbol{\textdollar}\textbackslashtextbackslashGamma \textdollar)\) Lacks Equivalent URED Norm

Assume the opposite: suppose there is an equivalent URED norm on \(c_0(\boldsymbol{\textdollar}\textbackslashtextbackslashGamma \textdollar)\). Consider canonical unit vectors: \(e_\textbackslash\textbackslashgamma\). Then investigate the bounded sequences fulfilling the URED condition. This can show contradictions with URED nature, revealing \(c_0(\boldsymbol{\textdollar}\textbackslashtextbackslashGamma \textdollar)\) can’t have an equivalent URED norm.

Key concepts

Banach space

A Banach space is a special type of vector space that incorporates a norm, which measures the 'size' of vectors. What makes a Banach space unique is that it is complete with respect to the distance defined by its norm. This means any Cauchy sequence (a sequence where the elements become arbitrarily close to each other as the sequence progresses) in the space has a limit that is also within the space.

Norms in Banach spaces help us understand the structure and behavior of sequences and functions within these spaces. The norm is usually denoted as \(orm{x}\) and must satisfy four properties: positivity, definiteness, homogeneity, and the triangle inequality.

Uniformly Rotund in Every Direction (URED)

A Banach space is said to be uniformly rotund in every direction (URED) if, roughly speaking, every direction experiences rotundness uniformly. This property implies that for every element in the unit sphere (elements of norm 1) and all bounded sequences \(\[x_n\], \[y_n\]\) with specific properties, a certain behavior or convergence is observed.

In more detail, if we have bounded sequences \(orm{x_n} = orm{y_n} = 1\) fulfilling:

\[2 orm{x_n}^2 + 2 orm{y_n}^2 - orm{x_n + y_n}^2 \rightarrow 0\]
and if the difference of these sequences in terms of another vector converges to zero, then the Banach space demonstrates this rotund property.

Unit sphere in Banach space

The unit sphere in a Banach space, denoted as \[S_X\], is the set of all vectors in the space that have a norm of 1. Formally,

\[S_X = \{ x \in X : orm{x} = 1 \} \]
Understanding the unit sphere is essential as it often forms the core of convergence and other properties studied in functional analysis.

The unit sphere helps identify how the space behaves in different directions and is critical when analyzing properties like the URED norm. For any sequence of points on this sphere, the behavior of these points under various mappings defines many of the space's structural characteristics.

Bounded sequences

A sequence \[x_n\] in a Banach space is said to be bounded if there exists a constant \(\text{M}\) such that for all \(\text{n}\), \( orm{x_n} \leq M \). This essentially means the elements of the sequence do not grow indefinitely.

Bounded sequences are crucial in the study of Banach spaces because they allow us to apply various limit theorems and convergence criteria. They play a significant role in the definition and properties of URED norms, as described above.

Equivalent norms in functional analysis

Two norms \(orm{\textunderscore}_1\) and \(orm{\textunderscore}_2\) on a vector space are called equivalent if there exist positive constants \(c_1\) and \(c_2\) such that for any vector \(\text{x}\),

\[c_1 orm{x}_1 \leq orm{x}_2 \leq c_2 orm{x}_1\]
This means that the different norms define the same topology, so their sequences and limits behave equivalently.

In the context of Banach spaces and URED norms, showing that no equivalent URED norm exists for certain spaces like \[c_0(\text{Gamma})\] can reveal deeper insights into the structure and limitations of these spaces.