# Complement a finite dimensional subspace in a Banach space

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?

Let $$e_1,..,e_n$$ be a basis of $$F$$, consider $$f_i$$ on $$vect(e_1,...,e_n)$$ by $$f_i(e_i)=1, f(e_j)=0, j\neq i$$, you can extend it to $$X$$ by Hahn Banach. Write $$W=\cap_{i=1,..,n}Ker g_i$$.
$$W$$ is closed, if $$x\in X, x=g_1(x)e_1+..+g_n(x)e_n+(x-g_1(x)e_1+..+g_n(x)e_n)$$,
write $$y=x-(g_1(x)e_1+..+g_n(x)e_n)$$, $$g_i(y)=0$$, and $$x\in Vect(e_1,...,e_n)\cap W$$ implies that $$x=x_1e_1+...+x_ne_n$$, and $$g(x)=x_i=0$$, we deduce that $$x=0$$.
• What is $g_i$? I guess you mean $f_i$? – Dave Shulman May 13 '19 at 17:01
• $f_i$ is defined on $Vect(e_1,...,e_n)$ one needs Hahn Banach to extend it to $X$ in $g_i$. – Tsemo Aristide May 13 '19 at 17:05