Proving the Continuity of Norms in Hilbert Spaces for q>=p

[MathematicalPhysicist]

Homework Statement

Prove that for q>=p and any f which is continuous in [a,b] then [tex]|| f ||_p<=c* || f ||_q[/tex], for some positive constant c.

Homework Equations

The norm is defined as: [tex]||f||_p=(\int_{a}^{b} f^p)^\frac{1}{p}[/tex].

The Attempt at a Solution

Well, I think that because f is continuous so are f^p and f^q are continuous and on a closed interval which means they get a maximum and a minimum in the interval which are both positive (cause if f were zero then the norm would be zero and the ineqaulity will be a triviality), so f^q>=M2, f^p<=M1, and we get that:
[tex]||f||_p/||f||_q<=M1^{1/p}/M2^{1/q}(b-a)^{1/p-1/q}[/tex] which is a constant.

QED, or not?

Thanks in advance.

 

[e(ho0n3]

Your norm is wrong. It should be

[tex]
\|f\|_p=\left(\int_{a}^{b} |f|^p\right)^\frac{1}{p}
[/tex]

This means that you need to argue using |f|, not f. Otherwise, the argument is OK.