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MathematicalPhysicist
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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.