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Let $X=C([0,1])$ and $T: X \rightarrow X$ defined as $$(Tf)(t)=f(t)+f(0)$$ Prove $T$ is bounded.

I was thinking about using the fundamental theorem of calculus in order to get some bounds on $f(0)$ but still don't manage to find the final result $||Tf|| \leq c ||f||$.

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1  
Which norm do you put on $X$? –  Davide Giraudo May 26 at 16:24
3  
Use: $|f(0)|\le\Vert f\Vert_\infty$. –  David Mitra May 26 at 16:26

1 Answer 1

What about $||Tf||_X=||Tf||_\infty=\sup_{t\in[0,1]}|f(t)+f(0)|\le 2||f||_\infty=2||f||_X$? Hence, $T$ is bounded.

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