Unitary representations of compact groups are discrete Hilbert sums of irreducibles

Let $\mathcal{S}$ be the set of all sets $\mathcal{F}$ of pairwise orthogonal nonzero closed finite-dimensional $\pi (K)$-invariant subspaces $M\subseteq H$ on which $\pi$ restricts irreducibly, ordered by inclusion; $\mathcal{S}$ is nonempty because by [F1] $H$ contains a nonzero finite-dimensional closed invariant subspace, which by [F2] contains a nonzero irreducible invariant subspace. Every chain in $\mathcal{S}$ has the upper bound $\bigcup \mathcal{C}$, which lies in $\mathcal{S}$ because any two of its members lie in a common member of the chain and are therefore orthogonal, while none of them is zero; hence by Zorn [F4] there is a maximal $\mathcal{M}\in \mathcal{S}$. Its members are pairwise orthogonal closed invariant subspaces, so their closed linear span $N$ is their Hilbert direct sum and is $\pi (K)$-invariant, being the closed span of invariant subspaces.

If $N\mathrm{\ne }H$, then $N^{\perp }\mathrm{\ne }\{0\}$ is a nonzero closed invariant subspace by [F3], and [F1] applied to $\pi {\mathrm{\mid }}_{N^{\perp }}$ provides a nonzero finite-dimensional closed invariant subspace $W\subseteq N^{\perp }$; by [F2] $W$ contains a nonzero irreducible invariant subspace $M_0$, which is orthogonal to every member of $\mathcal{M}$ because it lies in $N^{\perp }$, so $\mathcal{M}\cup \{M_0\}$ is a strictly larger member of $\mathcal{S}$, contradicting maximality; hence $N=H$, that is, $H$ is the Hilbert direct sum of the family $\mathcal{M}$. Each member $M\in \mathcal{M}$ is an irreducible representation of $K$, hence finite dimensional with class $\sigma (M)\in \widehat{K}$ by [F5]; writing ${\mathcal{M}}_{\sigma }:=\{M\in \mathcal{M}:\sigma (M)=\sigma \}$ and $H_{(\sigma )}:=\stackrel{}{\mathrm{span}}\bigcup {\mathcal{M}}_{\sigma }$, the $H_{(\sigma )}$ are pairwise orthogonal closed invariant subspaces with closed linear span $H$, so $H={\widehat{⨁}}_{\sigma \in \widehat{K}}{H}_{(\sigma )}$ and each nonzero $H_{(\sigma )}$ is the Hilbert direct sum of the copies ${\mathcal{M}}_{\sigma }$ of $\sigma$; this proves (1) and the first clause of (2), while the finite-dimensionality of every irreducible representation of $K$ is [F5].

The grouped subspace $H_{(\sigma )}$ from step 2.1 is contained in the closed span $H_{\sigma }$ of all $\sigma$-copies. Conversely, let $L$ be any $\sigma$-copy. For each $M\in \mathcal{M}$ of class $\tau \mathrm{\ne }\sigma$, the orthogonal projection $p_M$ commutes with $\pi (K)$, because $M$ and $M^{\perp }$ are invariant [F3]. Thus $p_M{\mathrm{\mid }}_L:L\to M$ is a bounded intertwiner between inequivalent irreducibles and is zero by [F8]. Hence $L$ is orthogonal to every such $M$, and therefore to their closed span. The orthogonal decomposition of step 2.1 implies that the complement of this closed span is exactly $H_{(\sigma )}$, so $L\subseteq H_{(\sigma )}$. Taking closed spans gives $H_{\sigma }=H_{(\sigma )}$. By [F6], $P_{\sigma }$ is the orthogonal projection onto this subspace, proving (3). AC is used in Zorn's lemma, the Haar-based projections and the cited suppliers.