  For $\alpha>0$, the Bargmann projection $P_\alpha$ is the orthogonal
projection from $L^2(\gamma_\alpha)$ onto the holomorphic subspace
$L^2_{hol}(\gamma_\alpha)$, where $\gamma_\alpha$ is the standard Gaussian
probability measure on $\C^n$ with variance $(2\alpha)^{-n}$. The space
$L^2_{hol}(\gamma_\alpha)$ is classically known as the Segal-Bargmann space. We
show that $P_\alpha$ extends to a bounded operator on $L^p(\gamma_{\alpha
p/2})$, and calculate the exact norm of this scaled $L^p$ Bargmann projection.
We use this to show that the dual space of the $L^p$-Segal-Bargmann space
$L^p_{hol}(\gamma_{\alpha p/2})$ is an $L^{p'}$ Segal-Bargmann space, but with
the Gaussian measure scaled differently: $(L^p_{hol}(\gamma_{\alpha p/2}))^*
\cong L^{p'}_{hol}(\gamma_{\alpha p'/2})$ (this was shown originally by Janson,
Peetre, and Rochberg). We show that the Bargmann projection controls this dual
isomorphism, and gives a dimension-independent estimate on one of the two
constants of equivalence of the norms.
