Question

Let \(p \in(1, \infty)\) and \(X_{n}\) be Banach spaces for \(n \in \mathbf{N}\). By \(X=\left(\sum X_{n}\right)_{p}\) we denote the normed linear space of all sequences \(x=\left\\{x_{i}\right\\}_{i=1}^{\infty}, x_{i} \in X_{i}\) such that \(\sum\left\|x_{i}\right\|_{X_{2}}^{p}<\infty\), with the norm \(\|x\|=\left(\sum\left\|x_{i}\right\|_{X_{\imath}}^{p}\right)^{\frac{1}{p}}\) Show that \(X\) is a Banach space and that \(X^{*}\) is isometric to \(\left(\sum X_{i}^{*}\right)_{q}\) (where \(\frac{1}{p}+\frac{1}{q}=1\) ) in the following sense: to \(f \in X^{*}\) we assign \(\left\\{f_{i}\right\\}_{i=1}^{\infty}\) such that \(f_{i} \in X_{i}^{*}\) and \(f\left(\left\\{x_{i}\right\\}_{i=1}^{\infty}\right)=\sum f_{i}\left(x_{i}\right)\) Note: This direct sum is sometimes also denoted \(\bigoplus_{p} X_{i}\). Hint: Follow the proof for \(\ell_{p}\), which is the case of \(X_{i}=\mathbf{R}\).

Short answer

The space \(X = \bigoplus_{p} X_{i}\) is a Banach space, and its dual space \(X^{*}\) is isometrically isomorphic to \( \bigoplus_{q} X_{i}^{*} \).

Step-by-step solution

Understand the Definition of the Space

The space \(X=\left(\sum X_{n}\right)_{p}\) consists of sequences \(x=\left\{x_{i}\right\}_{i=1}^{\infty}\), where each \(x_{i} \in X_{i}\) and \(\sum\left\|x_{i}\right\|_{X_{i}}^{p}<\infty\). The norm on this space is defined by \(\|x\|=\left(\sum\left\|x_{i}\right\|_{X_{i}}^{p}\right)^{\frac{1}{p}}\).

Use the Proof for \( \ell_{p} \) Spaces as a Guide

The problem suggests following the proof used for \( \ell_{p} \) spaces. For \( \ell_{p} \), the spaces \(X_{i}\) are just \(\mathbf{R}\). We'll adapt this approach to our situation where \(X_{i}\) are general Banach spaces.

Show Completeness

To show that \(X\) is a Banach space, start by assuming a Cauchy sequence \( \left\{ x^{(n)} \right\} \) in \(X\). This means, for every \( \epsilon > 0 \), there exists an \( N \in \mathbf{N} \) such that \( \| x^{(n)} - x^{(m)} \| < \epsilon \) for all \(n, m \ge N\). Since the spaces \( X_{i} \) are Banach, each \( x_{i}^{(n)} \in X_{i} \) forms a Cauchy sequence. Consequently, for each \(i\), there exists a limit \( x_{i} \in X_{i} \) such that \( x_{i}^{(n)} \rightarrow x_{i} \) in \( X_{i} \).

Verify the Cauchy Sequences Converge

Now, check if the sequence \( x = \left\{ x_{i} \right\} \) formed by limits is in \(X\). We need \( \sum \| x_{i} \|_{X_{i}}^{p} < \infty \). Use the fact that \( x^{(n)} \) being Cauchy implies the partial sums are bounded and use pointwise convergence, showing that \( x \in X \).

Establish the Norm Convergence

For each \( \epsilon > 0 \), choose \(N\) such that \( \| x^{(n)} - x \| < \epsilon \) for all \(n \ge N\). Since \( x^{(n)} \) converges to \( x \) in \(X\), conclude that the limit, \( x \), is indeed an element of \(X\). This proves that \(X\) is a Banach space.

Identify the Dual Space

Consider \( f \in X^{*} \). Define \( f_{i} \in X_{i}^{*} \) to be the restriction of \(f\) to the component \( X_{i} \). Check that \( f(x) = \sum f_{i}(x_{i}) \) makes sense and is bounded.

Use Hölder's Inequality

To show \(X^{*} \) is isometrically isomorphic to \( \left( \sum X_{i}^{*} \right)_{q} \), where \( \frac{1}{p} + \frac{1}{q} = 1 \), use Hölder's inequality: \( \left| \sum a_{i} b_{i} \right| \leq \left(\sum |a_{i}|^p\right)^{1/p} \left(\sum |b_{i}|^q\right)^{1/q} \). This ensures that the functional defined by \(\{f_{i}\}\) is in \(\left( \sum X_{i}^{*} \right)_{q}\) and the norm equality holds.

Prove the Norm Equality

Verify that \( \| f \| = \left( \sum \| f_{i} \|_{X_{i}^*}^{q} \right)^{1/q} \), completing the isometric isomorphism between \(X^{*} \) and \( \left( \sum X_{i}^{*} \right)_{q} \).

Key concepts

Normed Linear Space

A normed linear space, or normed vector space, is a vector space combined with a function called a norm. The norm assigns a non-negative length or size to each vector in the space. The formal definition is simple: given a vector space \(V\) over a field \(F\) (common examples include the real numbers \(\mathbb{R}\) or the complex numbers \(\mathbb{C}\)), a norm on \(V\) is a function \(\| \cdot \| : V \rightarrow \mathbb{R}\) with the following properties for all \(x, y \in V\) and \(\alpha \in F\):

• Positive definiteness: \(\|x\| \ge 0\), and \(\|x\| = 0\) if and only if \(x = 0\).
• Homogeneity: \(\| \alpha x \| = |\alpha| \| x \|\).
• Triangle inequality: \(\| x + y \| \le \| x \| + \| y \|\).

The norm gives us a way to talk about the length and distance in a vector space, turning it into a metric space.

Isometric Isomorphism

An isometric isomorphism is a special type of linear map between two normed spaces. It preserves both the vector space structure and the norm. More formally, if we have two normed spaces \(X\) and \(Y\), a linear map \(T : X \rightarrow Y\) is called an isometry if it preserves norms, i.e., \(\|Tx\|_Y = \|x\|_X\) for all \(x \in X\). If \(T\) is also bijective (one-to-one and onto), then \(T\) is an isometric isomorphism. This means \(X\) and \(Y\) are not only algebraically the same but their geometric structure is identical as well. An isometric isomorphism shows that one space can be perfectly restructured into another without losing any information about the lengths of vectors.

Hölder's Inequality

Hölder's inequality is a fundamental result in real and functional analysis. It is used extensively in the study of \(\ell_p\) spaces and integrable functions. The inequality states that for sequences or functions, it provides a bound on the sum or integral of products of elements. For sequences, if \(a_i\) and \(b_i\) are elements of \(\mathbb{R}\), and \(1 < p, q < \infty\) with \(1/p + 1/q = 1\), Hölder's inequality is
\[ \left| \sum_i a_i b_i \right| \le \left( \sum_i |a_i|^p \right)^{1/p} \left( \sum_i |b_i|^q \right)^{1/q} \]
This inequality assures that the product of two sequences (or functions in the integral form) will be bounded by the product of their norms. It is crucial when dealing with dual spaces and proving various inequalities in analysis.

Dual Space

The dual space of a normed vector space \(X\), denoted by \(X^*\) or sometimes \(X'\), consists of all continuous linear functionals on \(X\). If \(X\) is a Banach space, so is \(X^*\). For each \(f\) in \(X^*\) and \(x\) in \(X\), the value \(f(x)\) is a scalar. The norm of a functional \(f\) in \(X^*\) is given by \(\|f\| = \sup \{ |f(x)| : \|x\| \le 1 \}\). This norm makes \(X^*\) into a normed space itself. Dual spaces are essential in functional analysis because they allow us to apply geometric intuition to infinite-dimensional spaces. The dual space also plays a crucial part in representing spaces like in the exercise, showing \(X^*\) is isometric to another space.

Cauchy Sequence

A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Formally, a sequence \(\{x_n\}\) in a metric space \((X, d)\) is Cauchy if for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n \ge N\), \(d(x_m, x_n) < \epsilon\). In normed spaces, this definition translates to \(\| x_m - x_n \| < \epsilon\) for some sufficiently large \(m, n\). Cauchy sequences are fundamental because they are used to define completeness of a space. A space where every Cauchy sequence converges is called complete or a Banach space when dealing with normed spaces. This concept was used in the exercise to show that \(X\) is a Banach space.