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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 relativealignment 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 l 2 , x n y n n x n y n
    • V 2 , 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 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 additionalstructure. 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
    • l 2 : Subsets of shifted Kronecker delta functions S δ n k k
    • 2 : 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.

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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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