Chapter 19

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 \(\|x\|=\sup \left|x_{k}\right|\).

Let \(\mathrm{E}\) be an incomplete normed linear space, with completion \(\bar{E}\). Prove that the conjugate spaces \(\mathrm{E}^{*}\) and \((\bar{E})^{*}\) are isomorphic. Hint. Given any \(\mathbf{f} \in \mathrm{E}^{*}\), extend \(\mathbf{f}\) by continuity to a functional \(\bar{f} \in(\bar{E})^{*}\). Conversely, given any \(f \in(\bar{E})^{*}\), let \(\mathbf{f}\) be the restriction of \(\bar{f}\) to \(\mathrm{E}\), namely the functional \(\mathbf{f}(x)=f(x)\) for all \(x \in \mathrm{E}\). Show that \(f \leftrightarrow f\) is the desired isomorphism (with \(\|\mathbf{f}\|=\|f\|)\).

Let E be a topological linear space, and let \(\mathrm{b}^{*}\) be the strong topology in \(\mathrm{E}^{* *}\) and \(\pi\) the natural mapping of \(\mathrm{E}\) into \(E^{* *}\). Prove that \(\pi^{-1}\) is continuous. Hint. The topology \(b^{*}\) induces a topology \(\pi^{-1}\left(b^{*}\right)\) in the space \(E\), in which a set \(\mathrm{G} \subset \mathrm{E}\) is said to be open if its image \(\pi(G)\) is the intersection of \(\pi(E)\) with an open subset of \(\left(E^{* *}, \mathrm{~b}^{*}\right)\). Show that \(\pi^{-1}\left(b^{*}\right)\) is stronger than the original topolngy in \(\mathrm{F}\).

Prove that every closed subspace of a reflexive space is itself reflexive.