I got this problem in my homework. Let $X$ be a normed space, and let $X_0\subset X$ be a subspace.
Let $T$ be a continuous linear function from $X_0$ to $Y$, where $Y$ is an $n$-dimensional normed space.
Prove that $T$ can be extended to a continuous linear function T' on $X$ s.t. \|T'\|\le n\|T\|
In my attempt to solve this problem, I took an Auberbach basis $\{e_i\},\{e^i\}$ of $Y$. I assumed WLoG that $T$ is onto $Y$ and for each $e_i$ chose an $x_i\in X_0$ s.t. $Tx_i=e_i$.
I then examined $\operatorname{Span}\{x_i\}$, this is an $n$-dimensional subspace of $X$, and I can thus define a projection $P$ from $X$ to $\operatorname{Span}\{x_i\}$ such that $\|P\|\le n$. I get that the composition $T\circ P$ is bound as needed, satisfies all the conditions and identifies with $T$ on $\operatorname{Span}\{x_i\}$. However, I wasn't able to prove that implies that it identifies with $T$ on the entire $X_0$. Furthermore, I wasn't even able to convince myself that it's true.
Any suggestions?
Thanks in advance