Inner Product Space 2.13 2002/01/04 2005/09/30 20:32:49.368 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Adan Galvan jago@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu norm inner normed space orthogonal orthonormal product Cauchy-Schwarz This module introduces inner product space. Next we equip a normed vector space V with a notion of "direction". An inner product is a function ( : · · V V ) such that the following properties hold ( x y z x V y V z V and α α  ): x y y x x α y α x y ...implying that α x y α x y x y z x y x z x x 0 with equality iff x 0 In simple terms, the inner product measures the relative alignment between two vectors. Adding an inner product operation to a vector space yields an inner product space. Important examples include: V N , ≔ x y x y V N , ≔ x y x y V l2 , ≔ x n y n n x n y n V ℒ2 , ≔ f t g t t f t g t The inner products above are the "usual" choices for those spaces. The inner product naturally defines a norm: ≔ x x x though not every norm can be defined from an inner product. An example for inner product space ℒ 2 would be any norm ≔ f p t f t p such that p 2 . Thus, an inner product space can be considered as a normed vector space with additional structure. Assume, from now on, that we adopt the inner-product norm when given a choice. The Cauchy-Schwarz inequality says x y x y with equality iff α  x α y . When x y , the inner product can be used to define an "angle" between vectors: θ x y x y Vectors x and y are said to be orthogonal, denoted as ⊥ x y , when x y 0 . The Pythagorean theorem says: ⊥ x y x y 2 x 2 y 2 Vectors x and y are said to be orthonormal when ⊥ x y and x y 1 . ⊥ x S means ⊥ x y for all y S . S is an orthogonal set if ⊥ x y for all x y S s.t. x y . An orthogonal set S is an orthonormal set if x 1 for all x S . Some examples of orthonormal sets are 3 : S 1 0 0 0 1 0 N : Subsets of columns from unitary matrices l2 : Subsets of shifted Kronecker delta functions S δ n k k ℒ2 : S 1 T f t n T n for unit pulse f t u t u t T , unit step u t where in each case we assume the usual inner product.