How to prove the space of bounded linear functionals is complete?

Question

How to prove the space of bounded linear functionals in a linear norm space is complete?

I just have no idea how to use assumptions about $f_n$ to prove $f$ is bounded? I do it like this,

Main idea is since $f_n$ is Cauchy sequence exist N , when $n,m>N$,$||f_n-f_m||<\epsilon$

so $||fn-f||<\epsilon$

$f_n$ is bounded,$||fn||<C_n$,then $||f||<\epsilon+C_n$ ,also bounded.

Is this right?

Answer 1

Let $(X<||.||$) be a normed space over $K$ complete field. Let $f_n:X\to K$ be a Cauchy sequence, that is: $||f_n-f_m|| \to 0$ if $n,m\to\infty$.

Hence, for each $x$, $f_n(x)$ is a Cauchy sequence in $K$. $|f_n(x)-f_m(x)|\le ||f_n-f_m||\cdot ||x||$. So $f_n$ converges pointwise, say $f_n(x)\to f(x)$. This $f$ will be linear. And bounded, as $||f_n||\le ||f_n-f_m||+||f_m||$, hence $||f_n||$ is also Cauchy sequence in $\Bbb R^+$.