# Thread: Prove Lemma on the norms of a function.

1. ## Prove Lemma on the norms of a function.

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!

2. No, I don't think your approach to (i) works. You have not assumed that f is differentiable. However, f is a continuous function on a closed interval. What do you know about such functions? Incidentally, I think there's an error in the problem statement. It should have

$\displaystyle \displaystyle W=\left(\int_{a}^{b}w^{2}(x)\,dx\right)^{1/2}.$

As for (ii), for the first inequality, I would experiment with (constant) functions like

$\displaystyle f(x)=\dfrac{\epsilon}{2(b-a)}.$

You may have to take square roots, or squares, or something to get that to work with the 2-norm.

For (ii) on the second inequality, again, you're dealing with continuous functions on closed intervals. Just think about a constant function. Is there a constant function that might satisfy that inequality?

3. Originally Posted by Mollier
(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

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

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
You are making this too complicated. All you need is to use the fact that $\displaystyle |f(x)|\leqslant\|f\|_\infty$ for all x in [a,b]. The weighted L_2-norm is then given by

$\displaystyle \displaystyle \|f\|_2^2 = \int^b_a|f(x)|^2w(x)\,dx \leqslant \int^b_a\|f(x)\|_\infty^2w(x)\,dx=\|f(x)\|_\infty^ 2W^2.$

Now take square roots.

Originally Posted by Mollier
(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

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

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

...

(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...
That is correct. It is easiest to construct such a function on the unit interval, for example $\displaystyle f(x) = n^kx^n$, where n is large, and $\displaystyle n^k$ is some suitable power of n, probably $\displaystyle n^{1/2}$ or something like that. Once you have found a function that works on the unit interval, you can translate it to the interval [a,b], to get $\displaystyle f(x) = n^k\bigl(\frac{x-a}{b-a}\bigr)^n$

4. I do tend to overcomplicate, which I guess is fine if the result is correct...
Thanks.