So, the goal of the question was to prove that the following is a Banach Space:
$E:= \{a = (a_k)_{k=1}^\infty : \sum\limits_{k = 1}^{n}a_kx_k$ converges in $X\}$ equipped with the norm $||a||_E := \sup\limits_{n \in \mathbb{N}} ||\sum\limits_{k=1}^{n}a_kx_k||$. Where $(x_k)$ is a Schauder basis for a Banach space $X$.
Proving that the norm axioms hold is trivial, but I seem to be running into some trouble trying to prove that the space $E$ is complete. This is what I have, and I am pretty certain that it is not true:
Let ${y_n}$ be a Cauchy sequence in $E$ such that $y_n \rightarrow y$. $\sum\limits_{k=1}^{n}y_kx_k = \sum\limits_{k=1}^{n}(y_kx_k +yx_k - yx_k) = \sum\limits_{k=1}^{n}(y_k-y)x_k + \sum\limits_{k=1}^{n}yx_k$, which converges in $X$ so $y$ must be in $E$.
Any hints or encouragement would be welcomed. I am now considering a sequence of sequences.