Math 172: Inner Product Spaces, Symmetric Operators, Orthogonality

Definition 1. An inner product on a complex vector space V is a map 〈., .〉 : V × V → C such that (i) 〈., .〉 is linear in the first slot: 〈c1v1 + c2v2, w〉 = c1〈v1, w〉+ c2〈v2, w〉, c1, c2 ∈ C, v1, v2, w ∈ V, (ii) 〈., .〉 is Hermitian symmetric: 〈v, w〉 = 〈w, v〉, with the bar denoting complex conjugate, (iii) 〈., .〉 is positive definite: v ∈ V ⇒ 〈v, v〉 ≥ 0, and 〈v, v〉 = 0⇔ v = 0. A vector space with an inner product is also called an inner product space. While one should write (V, 〈.., .〉) to specify the inner product space, one typically says merely that V is an inner product space when the inner product is understood. For real vector spaces, one makes essentially the same definition, except that, as the complex conjugate does not make sense, one simply has symmetry: V real vector space ⇒ 〈v, w〉 = 〈w, v〉, v, w ∈ V. We also introduce the notation for the norm associated to this inner product: ‖v‖ = 〈v, v〉, where the square root is the unique non-negative square root of a non-negative number (see (iii)). Thus, 〈v, v〉 = ‖v‖. Recall that in general a norm is defined by: