Question

Let \(l_{n}\) be the normed linear space of all sequences \(x=\) \(\left(x_{1}, \ldots, x_{k}, \ldots\right)\) with norm $$ \|x\|=\left(\sum_{k=1}^{\infty}\left|x_{k}\right|^{p}\right)^{1 / p}<\infty \quad(p>1) $$ Prove that a) If \(\mathrm{p}>1\), the space \(l_{p}^{*}\) conjugate to \(l_{p}\) is isomorphic to the space \(l_{\sigma}\), where $$ \frac{1}{p}+\frac{1}{q}=1 $$ b) If \(p>1\), the general form of a continuous linear functional on \(l_{p}\) is $$ \tilde{f}(x)=\sum_{x=1}^{\infty} x_{k} f_{k} $$ wherex \(=\left(x_{1}, \ldots, x_{k}, \ldots\right) \in l_{p}, \mathbf{f}=\left(f_{1}, \ldots, f_{k}, \ldots\right) \in l_{q}\) c) If \(\mathrm{p}=1, l_{1}^{*}\) is isomorphic to the space \(m\) of all bounded sequences \(x=\left(\mathrm{x}_{n}, \ldots, x_{k}, \ldots\right)\) with norm \(\|x\|=\sup \left|x_{k}\right|\).

Short answer

The conjugate space \(l_{p}^{*}\) is \(l_{q}\), every functional on \(l_{p}\) has form \(\tilde{f}(x)=\sum x_{k} f_{k}\), and \(l_{1}^{*}\) is isomorphic to bounded sequences \(m\).

Step-by-step solution

Understand Isomorphism of Spaces

We need to show that if \(p>1\), the conjugate space \(l_{p}^{*}\) is isomorphic to \(l_{q}\). Here, \(q\) is defined by \(\frac{1}{p}+\frac{1}{q}=1\), meaning \(q\) is the Hölder conjugate of \(p\). This involves showing there is a bijective, norm-preserving linear map between these spaces.

Use Hölder's Inequality for Isomorphism

Given the definition of \(l_{p}\) and \(l_{q}\), we use Hölder's inequality: for sequences \(x\in l_{p}\) and \(y\in l_{q}\), \(\sum_{k=1}^{\infty} |x_{k}y_{k}| \leq \|x\|_{p}\|y\|_{q}\). This allows us to define a linear functional \(f(y) = \sum_{k=1}^{\infty} x_{k}y_{k}\) that maps elements from \(l_{p}\) to \(l_{q}\). The map is bijective and continuous.

General Form of Continuous Linear Functional

For part b, we express a continuous linear functional \(\tilde{f}\) on \(l_{p}\) using \(\tilde{f}(x) = \sum_{k=1}^{\infty} x_{k}f_{k}\), where \(f\) is a sequence in \(l_{q}\). This representation is possible due to the dual pairing and linearity of functional with respect to sequences \(x\in l_{p}\) and \(f\in l_{q}\).

Isomorphism of \(l_{1}^{*}\) and Space of Bounded Sequences

For part c, we need to identify \(l_{1}^{*}\). Analogous to how dual spaces are constructed, \(l_{1}^{*}\) can be shown to be isomorphic to the space \(m\), the space of all bounded sequences. In this context, whether a sequence's sum converges or if each element remains bounded determines membership. The norm on \(m\) is given by the supremum of absolute values over all elements in the sequence.

Key concepts

Isomorphism of Spaces

Isomorphism of spaces is a concept in functional analysis that describes a relationship between two vector spaces when there exists a bijective (one-to-one and onto), linear, and norm-preserving map between them. In simpler terms, it means we can map every element of one space to a unique element of the other space in a way that respects the structure and operations of the spaces.

If two spaces are isomorphic, they are fundamentally the same in terms of their vector space properties, even though they may appear different at first glance. They have the same algebraic and topological structure, meaning the operations and relationships between vectors within the spaces are preserved.

In the context of the given exercise, for a space like the normed linear space of sequences denoted by \( l_p \), its isomorphic counterpart when considering the dual space \( l_p^* \), turns out to be \( l_q \), with \( 1/p + 1/q = 1 \). This is an important result in functional analysis, showing that the dual space of \( l_p \) aligns with another sequence space depending on this relationship.

Hölder's Inequality

Hölder's inequality is a fundamental inequality in mathematical analysis. It is a tool used to handle integrals and sums. In particular, it provides a way of bounding sums of products of functions with different types of integrability.

The inequality states that for sequences \( x \in l_p \) and \( y \in l_q \), where \( 1/p + 1/q = 1 \), the sum of their products satisfies:

• \( \sum_{k=1}^{\infty} |x_k y_k| \leq \|x\|_p \|y\|_q \)

This means that the sum of the products of the sequences is always less than or equal to the product of their norms.

Hölder's inequality is particularly instrumental in proving statements about dual spaces and establishing isomorphisms as it ensures continuity and boundedness of functionals. It helps guarantee that mappings between spaces preserve their structure in continuous ways. Without such inequalities, many foundational proofs in real and functional analysis wouldn't hold.

Continuous Linear Functional

A continuous linear functional is a linear map from a vector space to its field of scalars that is also continuous with respect to the topology induced by the norm on the vector space. In other words, it obeys the rules of linearity and does not "tear apart" the space in terms of continuity.

For any vector space \( V \), a functional \( f: V \rightarrow \mathbb{R} \) or \( \mathbb{C} \) is linear if:

• \( f(x+y) = f(x) + f(y) \) for all \( x, y \in V \)
• \( f(\alpha x) = \alpha f(x) \) for all scalars \( \alpha \) and \( x \in V \)

Additionally, the continuity of a linear functional is automatically guaranteed if the space is finite-dimensional. In infinite-dimensional spaces, like \( l_p \), a functional such as \( \tilde{f}(x) = \sum_{k=1}^{\infty} x_k f_k \) uses sequences from \( l_q \) to maintain bounded, well-behaved behavior. Here, \( f_k \) comes from the dual relationship of these spaces, preserving the integral structure of \( l_p \) and \( l_q \).

Dual Spaces

Dual spaces extend the idea of pairing elements from a vector space with linear functionals. For a space \( V \), its dual space, denoted \( V^* \), consists of all possible continuous linear functionals that can act on \( V \). It's a way of analyzing space from the view of its functionals.

In the setting of sequence spaces like \( l_p \), the dual space \( l_p^* \) relates closely to other sequence spaces through the construction based on conjugate exponents. Specifically, the dual space representation is given by space \( l_q \), where \( 1/p + 1/q = 1 \). This isomorphism between \( l_p^* \) and \( l_q \) is vital in analysis because it allows us to leverage the properties of \( l_q \) in understanding \( l_p^* \).

When the original space \( V \) includes all bounded sequences, like \( l_1^* \) in our exercise, its dual space includes functional mappings over these sequences, broadened further by the space of all bounded sequences, \( m \). This reflects how duality captures both the algebraic structure and the infinite aspect of boundedness.