Let X be a normed space, and Y finite dimensional subspace.Prove that there exist closed subspace Z of X such that X=Z+Y(direct sum).

Question

I'm dealing with the next problem.

Let X be a normed space, and Y $\leq$ X, finite-dimensional subspace. Prove that there exist closed, subspace Z $\leq$ X such that holds Y$\overset{\cdot}{+}$Z=X.

So, Y must be a complete subspace, therefore closed. I was thinking to use the statement followed from Hahn-Banach theorem; for every $x\in X\setminus Y$ there exist linear functional $f\in X'$ such that $f|_Y=0$, $||f||=1$ and $f(x)=d(x,Y)$. So candidate for subspace Z could be $X\setminus Y$? But that set is clearly open so I don't think it would be a good candidate?
Any suggestion? Thanks.

Answer 1

Let $e_1,...,e_n$ be a basis of $Y$, consider $f_1,..,f_n$ the dual basis of $e_1,...,e_n$. Hahn Banach implies that you can extend $f_i$ to $g_i$. Let $Z=\bigcup_{i=1,..,n}Ker (g_i)$. $Z$ is closed.

For every $x\in X$, $x-g_1(x)e_1..-g_n(x)e_n$ is an element of $Z$. You also have $Y\cap Z=\{0\}$.