came across this problem in a book I'm reading:
Let $\mathbb{X}$ be a Banach space. For a sequence of nonzero vectors $(x_k)_{k=1}^{\infty}$ in $\mathbb{X}$, define $\mathbb{E}:=\{a=(a_k)_{k=1}^{\infty}:\sum_{k=1}^{\infty}a_kx_k \text{ converges in }\mathbb{X}\}$ and $||a||_{\mathbb{E}}:=\sup_{n\in\mathbb{N}}||\sum_{k=1}^{n}a_kx_k||$. Prove that $\mathbb{E}$ is a Banach space.
I am completely stuck, so much so that I don't even know where to begin. I know that the usual proof of completeness of a space relies on translating everything to the reals and using the property that reals are complete, but I don't even know how to do that here. Any help is appreciated. Thank you.
$$\def\ne{\mathbb E}$$ To sum up my comments and give you some more elements. Consider a Cauchy sequence $b^{(j)}$ in $\mathbb E$. Then, $\|b^{(j)}-b^{(l)}\|_\ne$ goes to zero for large $j,l$. For every $n$, one has $$\|\sum_{k=1}^n(b^{(j)}_k-b^{(l)}_k)x_k\|\leq \|b^{(j)}-b^{(l)}\|_\ne.$$ Considering two consecutive numbers $n-1$ and $n$, using the triangular inequality, one obtains
$$|b^{(j)}_n-b^{(j)}_n|\|x_n\|=\|\sum_{k=1}^n(b^{(j)}_k-b^{(l)}_k)x_k -\sum_{k=1}^{n-1}(b^{(j)}_k-b^{(l)}_k)x_k \|\leq$$ $$\leq \|\sum_{k=1}^n(b^{(j)}_k-b^{(l)}_k)x_k\| +\|\sum_{k=1}^{n-1}(b^{(j)}_k-b^{(l)}_k)x_k \|\leq 2 \|b^{(j)}-b^{(l)}\|_\ne.$$ Since $b^{(j)}$ is a Cauchy sequence, this shows that also $b^{(j)}_n$ is a Cauchy sequence for every $n$. Can you continue from here?