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I am an engineering student who has unwittingly taken a module in functional anylysis which, unfortunately, is ever so slightly over my head. I would greatly appreciate if you could either point me in the right direction for the questions below, or at least tell me where I might find some materials which would enlighten me in this area.

So, $ \varphi(f)=\int u\times f $$ u\in L^\infty, f\in L^1 $ Show that $\varphi(f)$ is a continuous linear form on $L^1$.

Now, if $ A=\text{sup}{ \{ |\varphi(f)| \text{ for } f\in L^1 \text{ and } ||f||_1 \leq1 \}} $ can $A=\infty$?

Many thanks.

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    Holder inequality (I imagine you could do it more directly using the above comment)2011-11-13

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To show that $\varphi$ is continuous, you have to show that $A$ is finite. Let $M=||u||_\infty$. Then for $||f||_1\le 1$: $|\varphi(f)|=\biggl|\int uf\,\biggr|\le\int|uf|\le \int M|f|=M\int |f|=M||f||_1\le M.$

That $\varphi$ is linear follows from the linearity of integration.

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    @NIM: The estimate is a bit less trivial, but works the same way: use [Hölder's inequality](http://en.wikipedia.org/wiki/Holder's_inequality).2011-11-13