Suppose that the functions are integrable on the closed, bounded interval [a,b]. Prove that:

.

Proof.

Now,

Choose

then the previous equation becomes

which implies

How do I get rid of the 2 at the bottom?

Thanks.

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- Jan 16th 2009, 12:24 PMtttcomraderCauchy-Schwarz inequality for Integrals
Suppose that the functions are integrable on the closed, bounded interval [a,b]. Prove that:

.

Proof.

Now,

Choose

then the previous equation becomes

which implies

How do I get rid of the 2 at the bottom?

Thanks. - Jan 16th 2009, 12:50 PMThePerfectHacker
The Cauchy-Schwarz inequality is a general inequality for inner product spaces.

Thus, what you wrote is a consequence of this general inequality found here*.

*)I am just a little worried because the space of all integral functions is not an inner product space with norm .

However, the proof on that page seems to work. - Jan 16th 2009, 11:34 PMMoo
Hello,

A more general view on this inequality is Hölder's inequality.

Cauchy-Schwarz is then a specific situation, with p=q=2.

Hölder's inequality - Wikipedia, the free encyclopedia