1. The problem statement, all variables and given/known data Show that if V is a real inner-product space, then the set of self-adjoint operators on V is a subspace of L(V). 2. Relevant equations 3. The attempt at a solution Let M be the matrix representing T. Since we are dealing with real numbers, and T is self-adjoint, T=T* so M=MT. Let u and w be vectors. Now <Mu, w>=<u, MTw>. Since M=MT, <u, MTw>=<u, Mw>. If M=MT, M is an nxn matrix. And since M and MT are nxn, M and MT map a vector from an n-dimensional space V back to an n-dimensional space V. Since nxn matrices can only multiply into n-tall vectors, u and w must be n-tall vectors. Therefore u and w must be in V. Since M and MT are nxn matrices and can map an n-tall vector from V to an n-dimensional space (V), the set of self adjoint operators (which M and M^T are a part of)are within the set of L(V).