produces another function . Here is an -dimensional representation, and is a possibly infinite-dimensional representation. A special case of this construction is when is a space of functions defined on the a linear group itself, viewed as a -dimensional manifold embedded in formula_206 (with the dimension of the matrices). This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. formula_158