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.