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If $X$ is a normed vector-space, then $X^\ast$ is the normed vector-space of bounded linear functionals $X \rightarrow \mathbf{R}$. Assume $1 < p, q < + \infty$ such that $\frac{1}{p} + \frac{1}{q} = 1$.

I want to show that the embedding $\iota : L^p \rightarrow (L^p)^{\ast \ast}$, $f \mapsto (x^\ast \mapsto x^\ast(f))$ is the same as the composition of the isomorphisms $L^p \simeq (L^q)^\ast \simeq (L^p)^{\ast \ast}$.

But I'm having difficulties with the isomorphism $(L^q)^\ast \simeq (L^p)^{\ast \ast}$, namely to write it down explicitly. Can anyone help me on that? Or is there maybe another way I should tackle this?

Thanks a lot!

Steven
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  • IIRC, the reflexivity of $L^p$ follows from Clarkson's inequalities, which show that it is uniformly convex. This is not the only way to establish the reflexivity of $L^p$, but it's good to know about. – felipeh Jan 05 '16 at 18:43

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Let us define the isomorphisms $\newcommand\dual[2]{\langle#1,#2\rangle}\newcommand\dualspace{^\star}\newcommand\d{\mathrm{d}}$ \begin{align*} D_p : L^p(\mu) \to L^q(\mu)\dualspace, &\quad \dual{D_p \, f}{g}_{L^q(\mu)} = \int_\Omega f \, g \,\d\mu \\ D_q : L^q(\mu) \to L^p(\mu)\dualspace, &\quad \dual{D_q \, g}{f}_{L^p(\mu)} = \int_\Omega f \, g \,\d\mu \end{align*} Then, the adjoint $D_q^\star$ is an isomorphism $L^p(\mu)^{\star\star} \to L^q(\mu)^\star$. Moreover, it is useful to have \begin{equation*} \dual{D_p \, f}{g}_{L^q(\mu)} = \int_\Omega f \, g \, \d\mu = \dual{D_q \, g}{f}_{L^p(\mu)}. \end{equation*} for $f \in L^p(\mu)$ and $g \in L^q(\mu)$.

I hope this helps to show that $D_q^{-\star} \circ D_p$ is the embedding from $L^p(\mu)$ to $L^p(\mu)^{\star\star}$.

gerw
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