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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 ( x , y , z ,xVyVzV: xVyVzV x y z x V y V z V and α ,α: α α α ):
1. x,y=y,x¯ x y y x
2. x,αy=α(x,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,y)R 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: xy:x+y2=x2+y2 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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