If $X$ is a normed space and $Y$ is a finite dimensional subspace, then there exists a continuous linear projection $P$ from $X$ to $Y$.
Our teacher gave us the instruction to use the following fact:
Let $x_1,\cdots,x_n$ in $X$ be linearly independent. Then there exist $x'_1,\cdots,x'_n$ in $X'$ such that $x'_k(x_r)= \delta_{kr}, 1 \le k,r \le n$.
How does one proceed with this assumption?
Thank you!
[Math] continuous projections to finite dimensional subspaces of normed spaces
normed-spaces
Best Answer
@Yuki gave you the answer.
Define $P:X\to X$ by $Px = \sum x_k'(x) x_k$. Show that $P$ is linear, continuous and $P P = P$.