Let $(V,{\Vert \cdot \Vert)_V})$ and $(W,{\Vert \cdot \Vert_W})$ be two normed vector spaces over $\mathbb{R}$ and $L(V,W)$ the set of all continous linear operators from $V$ to $W$. Assuming that $W$ with the induced metric is complete, I want to show that $L(V,W)$ with the usual operator norm (supremum definition) is complete.
So I have to show that every Cauchy sequence in $L(V,W)$ converges in $L(V,W)$. My approach is to consider a Cauchy sequence in $L(V,W)$ and then for each $x \in V$ construct a corresponding sequence in $W$ and show that this sequence is Cauchy. My problem is that I am stuck on proving that this constructed sequence is Cauchy.
Recall the property of the operator norm that $\|T(x)\| \leq \|T\|_{op} \|x\|$