Closed Linear Subspace of BFS $X$ .

Question

This is the definition which would be useful: Let X be a Bannach Function Space on $[0;1]$. Assume that the space $C([0;1])$ of continuous functions on $[0;1]$ is a closed linear subspace of $X$.

Then for every $f \in C([0;1])$ we have $$||f|||_X \leq ||f||_{C([0;1])} $$ .

As I read It's because $||f||_X \leq ||\chi_{\Omega}||_X||f||_{C(I)}$,

assuming that $||\chi_{\Omega}||_X = 1 $.

I don't understand why do we have this inequality : $||f||_X \leq ||\chi_{\Omega}||_X||f||_{C(I)}$ . ?

Any help would be appreciated.

Answer 1

You have

$$ |f|=|f \chi_{\Omega}|\leq ||f||_{C(I)}\chi_{\Omega} $$ So by properties (2) and (3)

$$ ||f||_X=|||f|||_X\leq||\chi_{\Omega}||_X||f||_{C(I)} $$