I just did an exercise that goes like this
Let $X$ be a normed vector space and $M$ a closed subspace such that $dim (X/M)<\infty$, show that there exists a closed subspace $N$ such that $X=M\bigoplus N$.
Now my touhgt processe was well we know that every subspace of a normed vector space has an algebraic complement $N$ such that $X=M\bigoplus N$, and since $N \cong X/M$ we will have that $N$ has finite dimension and so it will be closed. Now do we need the fact that $M$ is a closed subspace of $X$, am I forgetting of justifying something?