What definition gives $\hat{x}\in X^{**}$?

Question

Definition: Let $X$ be a topological vector space and let $x\in X$. Then $x$ defines a linear functional $\hat{x}$ on $X^*$ via $\hat{x}(f)=f(x)$ $(f\in X^*)$.

let $X$ be a normed space and let $x\in X$. I am trying to show that $\hat{x}\in X^{**}$. And my attempts are:

1. Let $f\in X^*$ and let $(f_i)$ be a net in $X^*$ with $f_i\overset{\|\cdot\|}{\longrightarrow} f$ in $X^*$. Then $$|\hat{x}(f_i)-\hat{x}(f)|=|f_i(x)-f(x)|=|(f_i-f)(x)|\leqslant\|f_i-f\|\cdot\|x\|\rightarrow0.$$ Thus $\hat{x}(f_i)\rightarrow\hat{x}(f)$. Hence, $\hat{x}$ is a continuous linear functional on $X^*$; that is, $\hat{x}\in X^{**}$.

2. Using a corollary of the Hahn-Banach Theorem, $$||\hat{x}||=\sup\{|\hat{x}(x^*)|:\|x^*\|\leqslant1\}=\sup\{|x^*(x)|:\|x^*\|\leqslant1\}=\|x\|.$$ Thus $\hat{x}\in X^{**}$.

But my professor mentioned I didn't need any proof at all. It follows immediately by using definition in functional anaylsis. But I don't know what definition gives $\hat{x}\in X^{**}$? Any helps will be appreciated!!

Answer 1

From the fact that $f$ is bounded, you have $$ |\hat x(f)|=|f(x)|\leq\|f\|\,\|x\|. $$ So $\hat x$ is bounded and $\|\hat x\|\leq\|x\|$.

Answer 2

$V^*$ is the dual space of $V,$ and the double dual space $V^{**}$ is isomorphic to $V$. Can you take it from here?

Edit: I'm assuming $dim(V)<\infty$ here.

Answer 3

You've correctly noticed that something needs to be shown, in particular that $\hat{x}$ is a continuous linear functional. So, you just need to show that $\hat{x}$ is a bounded linear functional, which is easy: take $\|f\|=1$, then $$|\hat{x}(f)| = |f(x)| \leq \|x\|$$ hence $\|\hat{x}\|:=\sup_{\|f\|=1} |\hat{x}(f)|\leq\|x\|$, so is bounded.

Notice that with Hahn-Banach, you can additionally show that $\|\hat{x}\| = \|x\|$ (as you did), but that's not necessary here.