I am trying to understand the following proof in "Compact Lie Groups" by Sepanski:

Here $V$ could be infinite-dimensional, $\pi$ denotes an irreducible representation of $G$ on the Hilbert space $E_\pi$, and $V_{[\pi]}$ denotes the $\pi$-isotypical component of $V$. At this point in the text, it has already been proved that $E_\pi$ is finite-dimensional, and that $\text{Hom}_G(E_\pi,V)$ is a Hilbert space (with inner product $(\cdot,\cdot)_{\text{Hom}}$ determined by $(T_1,T_2)_{\text{Hom}} I = T_2^*\circ T_1$).

My question concerns the use of $\hat\otimes$ in the decomposition. Since $E_\pi$ is finite-dimensional, I believe the completion $\text{Hom}_G(E_\pi,V) \hat\otimes E_\pi$ should agree with the algebraic tensor product $\text{Hom}_G(E_\pi,V) \otimes E_\pi$. Yet Sepanski seems to make a point of distinguishing between them in the proof, so I feel I might be missing something. Am I?



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