Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider $C[0,1]$, the vector space of continuous functions on the interval $[0,1]$ and consider the map $N:[0,1]\to\mathbb R$ where $$N(f)=\sup_{x\in[0,1]}x\cdot|f(x)|.$$

I was able to show the homogeneity and subadditivity of $N$, but I do not know how to prove that $f=0\iff N(f)=0$. I feel like this follows from continuity, but I do not know how to give a proper $\epsilon$-$\delta$ proof. Can someone help me out?

share|improve this question
add comment

1 Answer 1

up vote 3 down vote accepted

Assume that $N(f)=0$ for $f\colon [0,1]\to \Bbb R$ a continuous function. We get $xf(x)=0$ for all $x\in [0,1]$. In particular, for $x\neq 0$, this gives $f(x)=0$. Now, we have by continuity $$f(0)=\lim_{n\to +\infty}f(n^{-1})=0$$ as $f(n^{-1})=0$ for each integer $n$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.