# Homework Help: Norms in Hilbert Spaces.

1. Aug 16, 2009

### MathematicalPhysicist

1. The problem statement, all variables and given/known data
Prove that for q>=p and any f which is continuous in [a,b] then $$|| f ||_p<=c* || f ||_q$$, for some positive constant c.

2. Relevant equations
The norm is defined as: $$||f||_p=(\int_{a}^{b} f^p)^\frac{1}{p}$$.

3. 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:
$$||f||_p/||f||_q<=M1^{1/p}/M2^{1/q}(b-a)^{1/p-1/q}$$ which is a constant.

QED, or not?

$$\|f\|_p=\left(\int_{a}^{b} |f|^p\right)^\frac{1}{p}$$