Peter–Weyl theorem

The Peter–Weyl theorem says that, for a compact topological group, matrix coefficients of finite-dimensional irreducible unitary representations are dense in continuous functions and span square-integrable functions.

Matrix coefficient: A function on a group obtained by evaluating a linear functional on the image of a vector under a representation. These functions are the theorem’s basic building blocks.
Compact group: A topological group whose underlying space is compact. Compactness is the central hypothesis ensuring the stated decomposition.
Harmonic analysis on groups: The study of functions and operators on groups using representation theory and invariant integration. Peter–Weyl supplies the basic spectral expansion for compact groups.
Ferdinand Georg Frobenius: A German mathematician whose work helped establish the character theory of finite-group representations. His character theory supplied a finite-group precursor to the theorem’s representation expansions.
Plancherel theorem: A theorem extending the Fourier transform as an isometry between square-integrable function spaces. It plays a related decomposition role, especially for noncompact groups.
Haar measure: A translation-invariant measure on a locally compact group, unique up to scale when the group is compact. It defines the invariant integral and square-integrable space in the theorem.
Unitary representation: A group homomorphism into unitary operators on a complex inner-product space. The theorem uses finite-dimensional irreducible representations of this kind.
Fourier transform on finite groups: A decomposition of functions on a finite group into matrix coefficients of its irreducible representations. It is the finite-group counterpart of the theorem’s expansion.
Issai Schur: A German mathematician whose work on representations includes foundational orthogonality relations. Schur’s results underpin orthogonality among the coefficients used in the theorem.
Pontryagin duality: A duality theorem describing locally compact abelian groups through their continuous characters. For abelian groups, the dual objects are characters rather than general irreducible representations.

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Fourier analysis on finite groupsRelated: Its finite-group form guarantees that irreducible matrix coefficients provide a complete orthogonal expansion.