with Sei = λi ei, then so that f is strictly positive and tends to ∞ as |X| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad X and each non-zero eigenvalue occurs with its negative, since i ad X is a skew-adjoint operator on the compact real form . So f has a global minimum at Y say. This minimum is unique, because if Z were another then where X in is defined by the Cartan decomposition If fi is an orthonormal basis of eigenvectors of ad X with corresponding real eigenvalues μi, then Since the right hand side is a positive combination of exponentials, the real-valued function g is strictly convex if X ≠ 0, so has a unique minimum. On the other hand, it has local minima at t = 0 and t = 1, hence X = 0 and p = exp Y is the unique global minimum. By construction f(x) = f(σ(h)xh−1) for h in H, so that p = σ(h)ph−1 for h in H. Hence σ(h)= php−1. Consequently, if g = exp Y/2, gHg−1 is fixed by σ and therefore lies in K. Applications Representation theory Maximal compact subgroups play a basic role in the representation theory when G is not compact. In that case a maximal compact subgroup K is a compact Lie group (since a closed subgroup of a Lie group is a Lie group), for which the theory is easier. The operations relating the representation theories of G and K are restricting representations from G to K, and inducing representations from K to G, and these are quite well understood; their theory includes that of spherical functions. Topology The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup K. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact K and a Euclidean space – G = K × Rd – thus in particular K is a deformation retract of G, and is homotopy equivalent, and thus they have the same homotopy groups. Indeed, the inclusion and the deformation retraction are homotopy equivalences. For the general linear group, this decomposition is the QR decomposition, and the deformation retraction is the Gram-Schmidt process. For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of