Complement a finite dimensional subspace in a Banach space

Question

Given a Banach space $(X,\|,\|)$, and a finite dimensional subspace $F \subset X$, is it always possible to choose a closed linear complement to $F$. Explicitly, I mean to say, will there always exist a closed linear subspace $K \subset X$, such that
$$
X \simeq F \oplus W?
$$
If yes, then what is the easiest way to see that this is the case?

Answer 1

Let $e_1,…,e_n$ be a basis of $F$, consider $f_i$ on $\text{span}\{e_1,…,e_n\}$ by $f_i(e_i)=1, f_i(e_j)=0, j\neq i$, you can extend it to $X$ by Hahn-Banach. Write $W=\cap_{i=1,\dots,n}\ker g_i$.

$W$ is closed, if $x\in X$, $x=g_1(x)e_1+\dots+g_n(x)e_n+(x-g_1(x)e_1+\dots+g_n(x)e_n)$,

write $y=x-(g_1(x)e_1+\dots+g_n(x)e_n)$, $g_i(y)=0$, and $x\in \text{span}\{e_1,\dots,e_n\}\cap W$ implies that $x=x_1e_1+\dots+x_ne_n$, and $g_i(x)=x_i=0$, we deduce that $x=0$.