3 added 11 characters in body
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Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation of $SO(3)$ in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations, which, in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional of $SU(2)$ representations you know everything.

Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation of $SO(3)$ in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations, which in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional representations you know everything.

Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation of $SO(3)$ in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations which, in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional of $SU(2)$ representations you know everything.

2 added 11 characters in body
source | link

Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation of $SO(3)$ in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations, which in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional representations you know everything.

Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations, which in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional representations you know everything.

Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation of $SO(3)$ in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations, which in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional representations you know everything.

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source | link

Since $SO(3)$ is compact, in view of Peter-Weyl's theorem, every unitary strongly continuous representation in a Hilbert space is a direct sum (not a direct integral) of finite-dimensional irreducible representations, which in turn, are finite dimensional representations of $SU(2)$. So, once you know all finite dimensional representations you know everything.