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Inner Product Space

Module by: Phil Schniter. E-mail the author

Summary: This module introduces inner product space.

Next we equip a normed vector space VV with a notion of "direction".

  • An inner product is a function ( (· · ·:V×V)C : · · V V ) such that the following properties hold (   ,   xVyVzV    x y z x V y V z V and   ,   α    α α ):
    1. x · y=y · x* x y y x
    2. x · αy=αx · y x α y α x y ...implying that αx · y=α*x · y α x y α x y
    3. x · y+z=x · y+x · z x y z x y x z
    4. x · x0 x x 0 with equality iff x=0 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:
    1. V=RN V N , x · yxTy x y x y
    2. V=CN V N , x · yxHy x y x y
    3. V= l2 V l2 , xn · ynn=xn*yn x n y n n x n y n
    4. V= 2 V 2 , ft · gtft*gtdt 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: xx · x x x x though not every norm can be defined from an inner product. 1 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|xy x y x y with equality iff α:x=αy α x α y .

When x · yR x y , the inner product can be used to define an "angle" between vectors: cosθ=x · yxy θ x y x y

  • Vectors xx and yy are said to be orthogonal, denoted as xy x y , when x · y=0 x y 0 . The Pythagorean theorem says: x+y2=x2+y2  ,   xy    x y x y 2 x 2 y 2 Vectors xx and yy are said to be orthonormal when xy x y and x=y=1 x y 1 .
  • xS x S means xy x y for all yS y S . SS is an orthogonal set if xy x y for all xyS x y S s.t. xy x y . An orthogonal set SS is an orthonormal set if x=1 x 1 for all xS x S . Some examples of orthonormal sets are
    1. R3 3 : S=( 1 0 0 )( 0 1 0 ) S 1 0 0 0 1 0
    2. CN N : Subsets of columns from unitary matrices
    3. l2 l2 : Subsets of shifted Kronecker delta functions S δnk kZ S δ n k k
    4. 2 2 : S= 1TftnT nZ S 1 T f t n T n for unit pulse ft=ututT f t u t u t T , unit step ut u t
    where in each case we assume the usual inner product.

Footnotes

  1. An example for inner product space 2 2 would be any norm f|ft|pdtp f p t f t p such that p>2 p 2 .

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