Functional's Distance from Kernel

Asked May 30 '17 at 05:10

Active May 30 '17 at 05:43

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Let $X=\{f\in C[0,1] ; f(0)=0\}$, $M=\{f\in X ; \int_{t=0}^{1}f(t)dt=1\}$.

I have to prove that $\forall f\in X , ||f||_{\infty}=1$ exists $d(f,M)=|\int_{t=0}^{1}f(t)dt|$.

I proved $d(f,M)\geq |\int_{t=0}^{1}f(t)dt|$, but I don't know how to prove the second direction of the inequality. Any suggestions?

edited May 30 '17 at 05:43

asked May 30 '17 at 05:10

user

1

It is hard to parse your question. – copper.hat May 30 '17 at 05:27
Sorry, I forgot '' before the definition of M, I hope that it is clear now – user May 30 '17 at 05:43

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