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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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    There are some details missing for this problem, such as what space these functions $f$ and $u$ are $L^1$ and $L^{\infty}$ on exactly, and what the space of the codomains of these functions are. But I think this can get you started: A linear operator is bounded if and only if it is continuous. Thus, just show that $\varphi$ is a linear operator, and $ \| \varphi (f) \|_1 \leq M \| f \|_1$ for some $M >0.$2011-11-13
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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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