Conway Functional Analysis text example?

[xeno_gear]

Hello, I'm reading through John Conway's A Course in Functional Analysis and I'm having trouble understanding example 1.5 on page 168 (2nd edition):

Let (X, \Omega, \mu) and M_\phi : L^p(\mu) \to L^p(\mu) be as in Example III.2.2 (i.e., sigma-finite measure space and M_\phi f = \phi f is a multiplication operator, where \phi \in L^\infty(X, \Omega, \mu)). If 1 \le p < \infty and 1/p + 1/q = 1, then M_\phi^* : L^q(\mu) \to L^q(\mu) is given by M_\phi^*f = M_\phi f. That is, M_\phi^* = M_\phi.

I think I see how to do it if it's L^2 since there we have an inner product to work with. But otherwise I'm just not sure and don't really get the example. Any ideas?

 

[Landau]

The essense is that the dual of L^p is L^q, via

L^q\to (L^p)^*
g\mapsto (f\mapsto \int fg)

Use good old Hölder's inequality to show that this mapping makes sense. That this is an isomorphism is not trivial, but it is probably explained in your measure theory book.

Once you have proven (or accepted) this result, Conway's claim is not hard anymore; just write out the definitions. Of course, he is being a bit sloppy, piling identifications on top of each other (e.g. it doesn't make sense to say M*=M, as they have totally different domain and codomain).