Some Notes on the Spectral Theorem

The theory has one formulation in terms of real inner product spaces and one in terms of complex Hermitian spaces. In fact the real case is naturally encompassed by the complex case. But since the real case is so often used, we will give both formulations starting with the real case. Let V be a real vector space. A real pairing 〈·, ·〉 on V , 〈·, ·〉 : V × V → R, (~v, ~ w) 7→ 〈~v, ~ w〉 is symmetric if for every ~v, ~ w in V , 〈~ w,~v〉 = 〈~v, ~ w〉. A pairing which is symmetric turns out to be bilinear if and only if for every ~v1, ~v2, ~ w in V and for every scalar c in R, 〈c~v1 + ~v2, ~ w〉 = c · 〈~v1, ~ w〉+ 〈~v2, ~ w〉. For every ~v in V , define ‖~v‖ by ‖~v‖ := 〈~v,~v〉. A symmetric, bilinear pairing is positive definite if for every nonzero ~v in V , ‖~v‖ is a positive real number. In this case define ‖~v‖ by ‖~v‖ := √ ‖~v‖2 = √ 〈~v,~v〉. Definition 2.1. A real inner product space is a pair (V, 〈·, ·〉) of a real vector space V and a symmetric, bilinear, positive definite pairing 〈·, ·〉 on V .