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Let $\varphi$ be a plurisubharmonic function on a bounded open set $\Omega\subset\mathbb{C}^{n}$. For every $m>0$ m>0$ let $\mathcal{H}_{m\varphi}(\Omega)$ be the Hilbert space of holomorphic functions $f$ on $\Omega$ such that $\int_{\Omega}|f|^{2}e^{-2m\varphi}<\infty$ and let $\psi_{m}(z)=\frac{1}{2m}\text{log}\sum |g_{m,k}|^{2}$ where $(g_{m,k})$ is an orthonormal basis of $\mathcal{H}_{m\varphi}(\Omega)$.

I don't understand why $\sum |g_{m,k}|^{2}$ is the square of the norm of the evaluation linear form $f\rightarrow f(z)$ on $\mathcal{H}_{m\varphi}(\Omega)$. because Because if $f=\sum a_{i}g_{i}$ then $|f|^{2}=\sum a_{i}$ ? moreover Moreover why does the following holdshold: $\psi_{m}(z)=\text{sup}_{f\in B(1)]\frac{1}{m}\text{log}|f(z)|$ $\psi_{m}(z)=\text{sup}_{f\in B(1)}\frac{1}{m}\text{log}|f(z)|$$ where $B(1)$ is the unit ball of $\mathcal{H}_{m\varphi}(\Omega)$.

thanks a lot
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Let $\varphi$ be a plurisubharmonic function on a bounded open set $\Omega\subset\mathbb{C}^{n}$. For every $m>0$ let $\mathcal{H}_{m\varphi}(\Omega)$ be the Hilbert space of holomorphic functions $f$ on$\Omega$ such that $\int_{\Omega}|f|^{2}e^{-2m\varphi}<\infty$ and let $\psi_{m}(z)=\frac{1}{2m}\text{log}\sum |g_{m,k}|^{2}$ where $(g_{m,k})$ is an orthonormal basis of $\mathcal{H}_{m\varphi}(\Omega)$.

I don't understand why $\sum |g_{m,k}|^{2}$ is the square of the norm of the evaluation linear form $f\rightarrow f(z)$ on $\mathcal{H}{m\varphi}(\Omega)$. \mathcal{H}_{m\varphi}(\Omega)$. because if `$f=\sum a{i}g_{i}$ $f=\sum a_{i}g_{i}$ then $|f|^{2}=\sum a_{i}$ ? moreover why the following holds: $\psi_{m}(z)=\text{sup}_{f\in B(1)]\frac{1}{m}\text{log}|f(z)|$ where $B(1)$ is the unit ball of $\mathcal{H}_{m\varphi}(\Omega)$.\mathcal{H}_{m\varphi}(\Omega)$.

thanks a lot
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evaluation linear form

Let $\varphi$ be a plurisubharmonic function on a bounded open set $\Omega\subset\mathbb{C}^{n}$. For every $m>0$ let $\mathcal{H}_{m\varphi}(\Omega)$ be the Hilbert space of holomorphic functions $f$ on$\Omega$ such that $\int_{\Omega}|f|^{2}e^{-2m\varphi}<\infty$ and let $\psi_{m}(z)=\frac{1}{2m}\text{log}\sum |g_{m,k}|^{2}$ where $(g_{m,k})$ is an orthonormal basis of $\mathcal{H}_{m\varphi}(\Omega)$.

I don't understand why $\sum |g_{m,k}|^{2}$ is the square of the norm of the evaluation linear form $f\rightarrow f(z)$ on $\mathcal{H}{m\varphi}(\Omega)$. because if `$f=\sum a{i}g_{i}$ then $|f|^{2}=\sum a_{i}$ ? moreover why the following holds:

$\psi_{m}(z)=\text{sup}_{f\in B(1)]\frac{1}{m}\text{log}|f(z)|$ where $B(1)$ is the unit ball of $\mathcal{H}_{m\varphi}(\Omega)$.

thanks a lot