Finding functions $g_{n}$ with norm $1$ s.t $|\phi(g_{n})|\to||\phi||$ where $\phi(g)=\int_{0}^{1}fg$ for some fixed $f\in C([0,1])$

Question

I am trying to do the following exercise:

Let $f\in C([0,1])$ and define a functional on $C([0,1])$ by $$ \phi(g)=\int_{0}^{1}fg $$

Prove that $\phi$ is is linear and bounded and find functions $\{g_{n}\}$ s.t $$ ||g_{n}||=1,\,|\phi(g_{n})|\to||\phi|| $$

I have managed to prove that $\phi$ is linear and bounded by $||f||=\int_{0}^{1}|f|$.

Can someone please help me with finding the $g_{n}$'s ?

score 3 · Accepted Answer · answered Oct 01 '13 at 10:42

3

Hint: Find continuous $g_n$'s approximating the following noncontinuous function $$g(x):={\rm sgn}(f(x))\,, $$ where $\ \rm sgn\ $ dentoes the signum function, ${\rm sgn}(s)=|s|/s\,$ if $s\ne 0$ and ${\rm sgn}(0)=0$.

answered Oct 01 '13 at 10:42

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Finding functions $g_{n}$ with norm $1$ s.t $|\phi(g_{n})|\to||\phi||$ where $\phi(g)=\int_{0}^{1}fg$ for some fixed $f\in C([0,1])$

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