Let $E$ be a measurable set and $1 < p < \infty$. Let $T$ be a continuous linear functional on $L^p[a,b]$ and $K = \{ f \in L^p(E) \ | \ ||f||_p \leq 1\}$. Find a function $f_0 \in K$ for which $T(f_0) \leq T(f) \text{ for all } f \text{ in } K$.
I thought contradiction will be easy to use, but it did get me anywhere, any clues or solutions?
Continuity property of a bounded linear functional $T$: if $\{f_n\} \to f$ in $X$, then $\{T(f_n)\} \to T(f)$.
I was going to leave this as a comment, but then I realized it was going to be kind of long, so this is just an observation, rather than an answer.
I'm not sure if it is true: if such a function $f_0$ existed, then this would imply that $|T(f_0)|\geq|T(f)|$ for all $f\in K$, but by definition $$\|T\|\colon=\sup_{f\in K}|T(f)|=|T(f_0)|,$$ whence this means that the sup is actually a max, and I don't believe that to be true for an arbitrary functional on $L^P$? (Edit: by which I mean, it shouldn't be too hard to imagine a functional whose norm is not attained by some function?)