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 ...

A vector space over a field K (R or C) is a set X with operations vector addition and scalar multiplication satisfy properties in section 3.1. [1] An inner product space is a vector space X with inner product 〈·, ·〉 : X ×X → K satisfying • 〈x + y, z〉 = 〈x, z〉+〈y, z〉, • 〈αx, y〉 =α〈x, y〉, • 〈x, y〉 = 〈y, x〉, • 〈x, x〉 ≥ 0 with 〈x, x〉 = 0 ⇐⇒ x = 0. [2] An inner product induces a norm on X via ‖x‖ =p...