# Elementary relationships between finite-dimensional subspaces and finite-codimensional subspaces

I have an elementary question about finite-dimensional subspaces and finite-codimensional subspaces. This question may be known.

Question. Let $$U$$ be a finite-dimensional subspace of an infinite-dimensional Banach space $$X$$. Let $$\epsilon>0$$. Is there a finite-codimensional subspace $$V$$ of $$X$$ such that $$U\cap V=\{0\}$$ and $$\|u\|\leq (1+\epsilon)\|u+v\|$$ for every $$u\in U, v\in V$$ ?

• (Of course the extra condition on $U \cap V$ is unnecessary, since, if $u \in U \cap V$, then your inequality gives $\|u\| \le (1 + \epsilon)\|u - u\| = 0$. Also there's no need to specify infinite dimensionality, since, if $X$ is finite dimensional, then you may take $V = 0$.) May 18, 2020 at 8:35
• The condition $U\cap V=\{0\}$ is necessary, LSpice. May 18, 2020 at 8:44
• My inequality implies that $U\cap V=\{0\}$. You are right, LSpice. But how to prove the result? May 18, 2020 at 8:47

Let $$\epsilon>0$$ and let $$(u_{i})_{i=1}^{n}$$ be a $$\delta$$-net for $$S_{U}(\delta>0$$ will be specified later). For each $$i$$, choose $$x^{*}_{i}\in S_{X^{*}}$$ such that $$\langle x^{*}_{i},x_{i}\rangle=1$$. Let $$V=\cap_{i=1}^{n}Ker(x^{*}_{i})$$. Then $$V$$ is the required finite-codimensional subspace.