I think the solution to this question somehow involves Riesz Representation Theorem, but I don't see how to apply it.

Suppose $\{X,\mathcal{M},\mu\}$ is a $\sigma-$ finite measure space, $1\leq p<\infty$ and $\phi$ in a continuous linear functional on $L^{p}(\mu)$.

a) Prove that if $1<p$ then there is an element $f\in L^{p}(\mu)$ such that $\mid\mid f\mid\mid_{p}=1$ and $\phi(f)=\mid\mid\phi\mid\mid$. b) prove that a) need not be true when p=1

  • $\begingroup$ You know that $L^p(\mu)' \cong L^q(\mu)$, where $\frac{1}{p}+\frac{1}{q} = 1$, probably. From that representation of $\phi$, you can construct the desired $f$ for $1 < p < \infty$ (for $\phi = 0$, and arbitrary $f$ will do). $\endgroup$ – Daniel Fischer Mar 25 '14 at 10:30
  • $\begingroup$ how can I prove $\mid\mid f\mid\mid_{p}=1$? $\endgroup$ – Absar Ul Haq Mar 25 '14 at 10:34
  • $\begingroup$ You construct it so that it has norm $1$. $\endgroup$ – Daniel Fischer Mar 25 '14 at 10:37
  • $\begingroup$ yes,Please help me how can I construct it? $\endgroup$ – Absar Ul Haq Mar 25 '14 at 10:38
  • $\begingroup$ How does $\phi$ look? What does the representation theorem say about that? $\endgroup$ – Daniel Fischer Mar 25 '14 at 10:39

For $1 < p < \infty$: If $\phi$ is a continuous linear functional on $L^{p}(\mu)$, then there exists $g \in L^{q}(\mu)$ such that $$ \phi(f)=\int fg\,d\mu,\;\;\; \left[\int |g|^{q}\,d\mu\right]^{1/q} =\|\phi\|. $$ See if you can figure out how to choose $f \in L^{p}$ to make the first integral look like the second, knowing that $1/p+1/q=1$, or $q/p+1=q$. See where it leads.


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