Hi.

Problem:

(i)Suppose that the real-valued weight function $\displaystyle w$ is defined, continuous, positive and integrable on the interval $\displaystyle (a,b)$. Then, for any function $\displaystyle f\in C[a,b]$,

$\displaystyle ||f||_2 \leq W||f||_{\infty}$,

where

$\displaystyle W=\bigg[\int^b_aw(x)dx\bigg]^{1/2}$.

(ii)Given any two positive numbers $\displaystyle \epsilon$ (however small) and $\displaystyle M$ (however large), there exists a function $\displaystyle f\in C[a,b]$ such that

$\displaystyle ||f||_2 < \epsilon$,

$\displaystyle ||f||_{\infty}>M$.

Attempt:

(i)By using integration by parts I get,

$\displaystyle \begin{aligned}

\int^b_a|f(x)|^2w(x)dx=&\bigg[|f(x)|^2\int w(x)dx\bigg]^b_a-2\int^b_a|f(x)|f'(x)\bigg(\int w(x)dx\bigg)dx \\

\leq& \bigg[|f(x)|^2\int w(x)dx\bigg]^b_a \\

\leq& \bigg(\max_{x\in [a,b]}|f(x)|\bigg)^2\int^b_a w(x)dx \\

\Rightarrow \bigg(\int^b_a|f(x)|^2w(x)dx\bigg)^{1/2}\leq& \max_{x\in [a,b]}|f(x)| \bigg(\int^b_a w(x)dx\bigg)^{1/2} \\

\Rightarrow ||f||_2 \leq& W||f||_{\infty}

\end{aligned}$

To me that looks good, but I've been mistaken before

(ii)I do not see how I can prove existence of the function. This somehow reminds me of the dirac delta function, as I can make my function as pointy as I want...

The book continues by saying that the Lemma indicates that the norms are not equivalent as they would be if this was a finite-dimensional linear space and we were dealing with vectors. That is if $\displaystyle ||*||_{\infty}$ and $\displaystyle ||*||_2$ are vector norms on $\displaystyle \mathbb{R}^n$ then

$\displaystyle n^{-1/2}||v||_{\infty}\leq ||v||_2 \leq n^{1/2}||v||_{\infty}, \forall v\in\mathbb{R}^n.$

Does this mean that if we had $\displaystyle W^{-1}||f||_{\infty} \leq ||f||_2$, then we could use the two norms interchangeably? I do not immediately see how the Lemma indicates that this is not so.

Thanks!