Reflexivity of L^p and its Implications for Integration

[Goklayeh]

Could someone confirm or refute the following statement?

$$f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right)$$

where $$1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1$$ and $$(X, \mu)$$ is a measurable space (of course, the $$(\Rightarrow)$$ is trivial by Holder inequality)

Thanks in advance!

 

[mathman]

It looks correct to me. From my recollection. L^p and L^q are adjoint, when p, q > 1 and 1/p + 1/q = 1.

 

[JSuarez]

For those values of p, L^p is reflexive. What can you infer from this?