Problem of dual space from krieszig functional analysis

Question

I am trying problems of functional analysis from krieszig book and I could not solve this problem from section 3.8 .

Problem is - Show that dual space of real space $ l^2 $ is $ l^2 $ . And he gives hint to use Riesz theoram ie every bounded linear functional f on a Hilbert Space can be represented in terms of inner product ie f(x) = , z depends on f.

Can someone please write a detailed argument, I am not good in problems of dual space. It will help me a lot to how to think about such problems.

Answer 1

You could start by showing that, given $y = (y_{n}) \in l^{2}$ you can define $T_{y}(x) = \sum_{n=1}^{\infty}{x_{n}\overline{y_{n}}}$ and $T_{y}$ is a continuous linear functional (you just have to use Holder's Inequality). We then identify $T_{y}$ and $y$ and say that $l^{2} \subset (l^{2})^{*}$. To prove that $(l^{2})^{*} \subset (l^{2})$ you can proceed by using Riesz Theorem: First note that the inner product in $(l^{2})$ is $<x, y> = \sum_{j=1}^{\infty}{x_{j}\overline{y_{j}}}$, then if you get $T \in (l^{2})^{*}$ a continuous linear functional, there exists $y$ such that $T(x) = <x, y>$ for every $x \in l^{2}$. Hence you can identify $T$ with $y$. These identifications define an isomorphism between $l^{2}$ and $(l^{2})^{*}$, so we say these spaces are equal.

Answer 2

the Riesz theorem says that fo each Hilbert space $H$ there is an antilinear bijection $f:H \rightarrow H^*$, $f(x) = \langle\cdot,x\rangle$. I.e. the dual space of a Hilbert space $H$ is just the Hilbert space $H$ itself, acting as the inner product. Hence, as $\ell_2$ is a Hilbert space, we get $\ell_2^* = \ell_2$.