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A [[Homomorphism|homomorphism]] of a [[Compact group|compact group]] into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operator topology.

Let $k$ be a compact group, let $V$ be a Banach space and let $\phi : G \rightarrow \operatorname { GL } ( V )$ be a representation. If $V = H$ is a Hilbert space and $\phi ( g )$ is a unitary operator for every $g \in G$, then $( 1 )$ is called a [[Unitary representation|unitary representation]]. There always is an equivalent norm in $H$ for which $( 1 )$ is unitary.

Every irreducible unitary representation (cf. [[Irreducible representation|Irreducible representation]]) of a compact group $k$ is finite-dimensional. Let $\{ \rho ^ { \alpha } : \alpha \in I \}$ be the family of all possible pairwise inequivalent irreducible unitary representations of the group $k$. Every unitary representation $( 1 )$ of $k$ is an orthogonal sum of unique representations $\phi ^ { a }$, $\alpha \in I$, such that $\phi ^ { a }$ is an orthogonal sum, possibly zero, of a set of representations equivalent to $\rho ^ { \alpha }$.

If $k$ is finite, then the family $\{ \rho ^ { \alpha } \}$ is also finite and contains as many elements as there are distinct conjugacy classes in $k$ (moreover, $\sum _ { \alpha \in I } ( \operatorname { dim } \rho ^ { \alpha } ) ^ { 2 } = | G |$). The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory (cf. [[Finite group, representation of a|Finite group, representation of a]]).

If $k$ is a connected, simply-connected, compact Lie group and $G _ { C }$ is its complexification (cf. [[Complexification of a Lie group|Complexification of a Lie group]]), then the description of the family $\{ \rho ^ { \alpha } : \alpha \in I \}$ for $k$ amounts (by restricting the representations to $k$) to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group $G _ { C }$. The latter family, in turn, allows of a complete description by considering highest weights (cf. [[Representation with a highest weight vector|Representation with a highest weight vector]]).

In modern number theory and algebraic geometry one considers $1$-adic representations of compact totally-disconnected groups (cf. [[#References|[5]]], [[#References|[6]]]).

====References====
<table><tr><td valign="top">[1]</td> <td valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </td></tr><tr><td valign="top">[2]</td> <td valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </td></tr><tr><td valign="top">[3]</td> <td valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </td></tr><tr><td valign="top">[4]</td> <td valign="top"> S. Lang, "$SL _ { 2 } ( R )$" , Addison-Wesley (1975)</td></tr><tr><td valign="top">[5]</td> <td valign="top"> I.M. Gel'fand, M.I. Graev, I.I. Pyatetskii-Shapiro, "Generalized functions" , '''6. Representation theory and automorphic functions''' , Saunders (1969) (Translated from Russian) {{MR|}} {{ZBL|0801.33020}} {{ZBL|0699.33012}} {{ZBL|0159.18301}} {{ZBL|0355.46017}} {{ZBL|0144.17202}} {{ZBL|0115.33101}} {{ZBL|0108.29601}} </td></tr><tr><td valign="top">[6]</td> <td valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0181643}} {{ZBL|0143.05901}} {{ZBL|0128.26303}} </td></tr><tr><td valign="top">[7]</td> <td valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946) {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </td></tr></table>



====Comments====


====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , ''Eléments de mathématiques'' , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts {{MR|0682756}} {{ZBL|0505.22006}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , '''II''' , Springer (1970) {{MR|0262773}} {{ZBL|0213.40103}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A. Wawrzyńczyk, "Group representations and special functions" , Reidel &amp; PWN (1984) {{MR|0750113}} {{ZBL|0545.43001}} </td></tr></table>