$L^p $ Spaces

[Valerio Capraro]

It's not easy!!

I'd like to know why (and where!) my proof is not correct ...

[Luca.Lussardi]

I don't know if it is correct or not... but why the map that closes the diagram should be the identity??

[Valerio Capraro]

I don't know how to insert a diagram ...
However:

for all $p\in[1,\infty)$ we have the following duality

$S : (L^p)$*$->L^{(1-1/p)^{-1}}$
$F->f:F(h)=\intfhdx, \forall h\inL^p$

I suppose now that $L^p$ is an Hilbert space. Thus there is the Riesz
represantion $R$

$R : (L^p)$*$->L^p$
$F->g:F(h)=\intgh, \forall h\inL^p$

I observe that $R(F)=S(F),\forall F\inL^p$*. Infact, the condition
$\intgh=\intfh,\forall h\in L^p$ leads to $f=g$ (for example we can consider
only test functions $h$ and apply the well-knowed theorem on test functions).
So, I can prove that $SR^{-1}=I$ and $RS^{-1}=I$. This proves that the
diagram commutes with $I$. This forces $p=(1-1/p)^{-1}$.

Now I'm quite sure that it is correct ... but maybe I am wrong!

[Luca.Lussardi]

I think there is a mistake in your proof: why $\int fg dx$ is a scalar product in $L^p$? This forces $p=2$, and the proof is essentially Camillo's proof.

[Valerio Capraro]

Ok! You're right! I used the thesis for proving them!
A bad mistake.