Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 22F / b / Solution

The continuous dual space of a normed vector space $V$ iswith the operator norm

Let $(f_n)$ be a Cauchy sequence in this norm. For each $v\in V$,so $(f_n(v))$ is Cauchy in the scalar field. Define $f(v)={\lim }_n{f}_n(v)$. Passing to limits in the linearity identities shows that $f$ is linear.

A norm-Cauchy sequence is bounded, say $\Vert f_n\Vert \le M$. Hence $|f(v)|\le M\Vert v\Vert$, so $f\in V^{\ast }$. Given $\epsilon >0$, choose $N$ such that $\Vert f_n-f_m\Vert <\epsilon$ for $m,n\ge N$. Letting $m\to \infty$ givesfor every $v$, and therefore $\Vert f_n-f\Vert \le \epsilon$. Thus $f_n\to f$ in operator norm, proving completeness of the dual space: $V^{\ast }$ is Banach.