I am thinking in this problem:
If $E$ is a normed space and $M\subseteq E$ is a subspace of finite dimension, prove that for all $x\in E-M$ there exists $m_0\in M$ such that $d(x,M)=\|x-m_0\|$.
I am trying to apply Riesz's Theorem because $M$ is a closed space (finite dimensional), but I don't know how….
Best Answer
By definition of distance, there exists $\{m_n\}\subset M$ such that $\|x-m_n\|\to d(x,M)$. From the triangle inequality, $$ \|m_n\|\leq\|x-m_n\|+\|x\|. $$ So the sequence $\{m_n\}$ is bounded. Because $M$ is finite-dimensional, it is closed, and also closed balls are compact. So there exists a convergent subsequence $m_{n_k}$. Let $m_0=\lim_k m_{n_k}\in M$. And we have $$ \|x-m_0\|=\lim_n\|x-m_n\|=d(x,M). $$