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Chapter 2: Problem 17

Let \(Y\) be a subspace of a Banach space \(X\). Show that there exists a one-to- one (in general, nonlinear) isometric map \(\varphi: Y^{*} \rightarrow X^{*}\). Also, \(\left.X^{*}\right|_{Y}=Y^{*}\left(\left.X^{*}\right|_{Y}\right.\) is the set of restrictions to \(Y\) of all \(\left.f \in X^{*}\right)\). Hint: Use the Hahn-Banach theorem to extend functionals on \(Y\).

Short Answer

Expert verified
Define a map \(\varphi\) from \(Y^{*}\) to \(X^{*}\) using the Hahn-Banach extension. This isometric map \(\varphi\) is both injective and retains the norm of the functionals.

Step by step solution

01

Define the functionals

Given a Banach space \(X\) and its subspace \(Y\), define a functional \(f \in Y^{*}\). Our goal is to extend \(f\) to a functional \(\varphi(f) \in X^{*}\) which retains the norm.
02

Apply the Hahn-Banach theorem

Use the Hahn-Banach theorem to extend \(f\). The Hahn-Banach theorem states that any bounded linear functional defined on a subspace of a normed space can be extended to the whole space with the same norm. Therefore, there exists an extension \(\tilde{f} \in X^{*}\) such that \(\tilde{f}|_{Y} = f\) and \(\|\tilde{f}\| = \|f\|\).
03

Define the isometric map

Define the map \(\varphi: Y^{*} \rightarrow X^{*}\) by letting \(\varphi(f) = \tilde{f}\), where \(\tilde{f}\) is the Hahn-Banach extension of \(f\).
04

Verify isometry and injectivity

To show that \(\varphi\) is isometric, note that for any \(f \in Y^{*}\), \(\|\tilde{f}\| = \|f\|\) by construction, hence \(\|\tilde{f}\| = \|\tilde{f}|_Y\|\). To show injectivity, observe that if \(\varphi(f_1) = \varphi(f_2)\), then \(\tilde{f}_1 = \tilde{f}_2\), implying \(f_1 = f_2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Banach space
A Banach space is a type of normed space that is complete, meaning every Cauchy sequence in the space converges to a point within the space. This is crucial because it ensures stability and predictability in the behavior of sequences within the space. Banach spaces are widely used in functional analysis and various branches of mathematics. They generalize many properties of Euclidean spaces. For instance, common examples of Banach spaces include the spaces of continuous functions, square-integrable functions, and all bounded sequences.
Linear functional
A linear functional is a linear map from a vector space to its field of scalars. In more practical terms, if you have a vector space of functions, a linear functional assigns a single scalar value to each function in a way that respects addition and scalar multiplication. Linear functionals are super useful tools in analysis because they help in understanding the structure of vector spaces. They often appear when dealing with dual spaces, where the set of all linear functionals itself forms a vector space. In our exercise, they are essential for extending a function from a subspace using the Hahn-Banach theorem.
Isometric map
An isometric map is a function between metric spaces that preserves distances. This means that the distance between any two points in the Domain maps exactly to the same distance in the Codomain. In our context, \(\tilde{f}\) is an isometric map ensuring that the norm is retained during the extension. Think of it as a way to keep the 'shape' of the space intact when mappings occur. By using an isometric map, we ensure that all properties related to distance and norms are preserved.
Normed space
A normed space is a vector space equipped with a function called a 'norm,' which assigns a strictly positive length or size to each vector in the space except for the zero vector. In formula terms, if \(\boldsymbol{x}\) is a vector in the space, its norm \(\boldsymbol{\text{||}\text{X}\text{||}}\) measures the 'distance' from \(\boldsymbol{0}\) to \(\boldsymbol{x}\). Norms help define notions of distance and convergence, laying the groundwork for more advanced concepts such as Banach spaces. Normed spaces are essential in functional analysis and various applications across physics and engineering because they provide the first step towards generalizing the idea of length from familiar contexts like Euclidean space.

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Most popular questions from this chapter

Show that there is no \(T \in \mathcal{B}\left(\ell_{2}, \ell_{1}\right)\) such that \(T\) is an onto map. Hint: \(T^{*}\) would be an isomorphism of \(\ell_{\infty}\) into \(\ell_{2}\), which is impossible since \(\ell_{\infty}\) is nonseparable and \(\ell_{2}\) is separable.

Let \(X, Y\) be Banach spaces and \(T \in \mathcal{B}(X, Y)\). If \(T\) is an isomorphism into \(Y\), is \(T^{*}\) necessarily an isomorphism into \(X^{*}\) ? Hint: No, embed \(\mathbf{R}\) into \(\mathbf{R}^{2}\).

Let \(X=\mathbf{R}^{2}\) with the norm \(\|x\|=\left(\left|x_{1}\right|^{4}+\left|x_{2}\right|^{4}\right)^{\frac{1}{4}} .\) Calculate directly the dual norm on \(X^{*}\) using the Lagrange multipliers. Hint: The dual norm of \((a, b) \in X^{*}\) is \(\sup \left\\{a x_{1}+b x_{2} ; x_{1}^{4}+x_{2}^{4}=1\right\\} .\) Define \(F\left(x_{1}, x_{2}, \lambda\right)=a x_{1}+b x_{2}-\lambda\left(x_{1}^{4}+x_{2}^{4}-1\right)\) and multiply by \(x_{1}\) and \(x_{2}\), respectively, the equations you get from \(\frac{\partial F}{\partial x_{1}}=0\) and \(\frac{\partial F}{\partial x_{2}}=0\)

Let \(X\) be a Banach space. (i) Show that in \(X^{*}\) we have \(X^{\perp}=\\{0\\}\) and \(\\{0\\}^{\perp}=X^{*}\). Show that in \(X\) we have \(\left(X^{*}\right)_{\perp}=\\{0\\}\) and \(\\{0\\}_{\perp}=X\). (ii) Let \(A \subset B\) be subsets of \(X\). Show that \(B^{\perp}\) is a subspace of \(A^{\perp}\). Hint: Follows from the definition.

Let \((X,\|\cdot\|)\) be a Banach space. Show that \(\mu_{B_{X}}(x)=\|x\|\). Hint: Use continuity of the norm.
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