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Hilbert Spaces

Module by: Phil Schniter. E-mail the author

Summary: This module introduces Hilbert spaces.

Now we consider inner product spaces with nice convergence properties that allow us to define countably-infinite orthonormal bases.

  • A Hilbert space is a complete inner product space. A complete 1 space is one where all Cauchy sequences converge to some vector within the space. For sequence xn xn to be Cauchy, the distance between its elements must eventually become arbitrarily small: ε ,ε>0: N ε : n , m ,(n N ε )(m N ε ): x n x m <ε ε ε 0 N ε n m n N ε m N ε x n x m ε For a sequence xn xn to be convergent to x, the distance between its elements and xx must eventually become arbitrarily small: ε ,ε>0: N ε : n ,n N ε : x n x<ε ε ε 0 N ε n n N ε x n x ε Examples are listed below (assuming the usual inner products):
    1. V=RN V N
    2. V=CN V N
    3. V= l 2 V l 2 (i.e., square summable sequences)
    4. V= 2 V 2 (i.e., square integrable functions)
  • We will always deal with separable Hilbert spaces, which are those that have a countable 2 orthonormal (ON) basis. A countable orthonormal basis for VV is a countable orthonormal set S= x k S x k such that every vector in VV can be represented as a linear combination of elements in SS: y,yV: α k :y=k α k x k y y V α k y k k α k x k Due to the orthonormality of SS, the basis coefficients are given by α k = x k ,y α k x k y We can see this via: x k ,y= x k ,limit   n i =0n α i x i =limit   n x k , i =0n α i x i =limit   n i =0n α i ( x k , x i )= αk x k y x k n i 0 n α i x i n x k i 0 n α i x i n i 0 n α i x k x i αk where δki= x k , x i δ k i x k x i (where the second equality invokes the continuity of the inner product). In finite nn-dimensional spaces (e.g., Rn n or Cn n ), any nn-element ON set constitutes an ON basis. In infinite-dimensional spaces, we have the following equivalences:
    1. x 0 x 1 x 2 x 0 x 1 x 2 is an ON basis
    2. If x i ,y=0 x i y 0 for all ii, then y=0 y 0
    3. y,yV:y2=i| x i ,y|2 y y V y 2 i i x i y 2 (Parseval's theorem)
    4. Every yV y V is a limit of a sequence of vectors in span x 0 x 1 x 2 span x 0 x 1 x 2
    Examples of countable ON bases for various Hilbert spaces include:
    1. Rn n : e0 e N - 1 e0 e N - 1 for ei =( 00100 )T ei 0 0 1 0 0 with "1" in the ith ith position
    2. Cn n : same as Rn n
    3. l2 l2: δi n iZ δi n i , for δi nδni δi n δ n i (all shifts of the Kronecker sequence)
    4. 2 2: to be constructed using wavelets ...
  • Say SS is a subspace of Hilbert space VV. The orthogonal complement of S in V, denoted S S, is the subspace defined by the set xV xS x V x S . When SS is closed, we can write V=SS V S S
  • The orthogonal projection of y onto S, where SS is a closed subspace of VV, is y ̂ = i i( x i ,y) x i y ̂ i i x i y x i s.t. x i x i is an ON basis for SS. Orthogonal projection yields the best approximation of yy in SS: y ̂ =argmin xS yx y ̂ argmin x S y x The approximation error ey y ̂ e y y ̂ obeys the orthogonality principle: eS e S We illustrate this concept using V=R3 V 3 (Figure 1) but stress that the same geometrical interpretation applies to any Hilbert space.
Figure 1
Figure 1 (projection1.png)

A proof of the orthogonality principle is: eS i :e, x i =0 e S i e x i 0 y y ̂ , x i =0 y y ̂ x i 0

y, x i = y ̂ , x i = j j( x j ,y) x j , x i = j j x j ,y¯( x j , x i )= j j(y, x j ) δ i j =y, x i y x i y ̂ x i j j x j y x j x i j j x j y x j x i j j y x j δ i j y x i
(1)

Footnotes

  1. The rational numbers provide an example of an incomplete set. We know that it is possible to construct a sequence of rational numbers which approximate an irrational number arbitrarily closely. It is easy to see that such a sequence will be Cauchy. However, the sequence will not converge to any rational number, and so the rationals cannot be complete.
  2. A countable set is a set with at most a countably-infinite number of elements. Finite sets are countable, as are any sets whose elements can be organized into an infinite list. Continuums (e.g., intervals of R) are uncountably infinite.

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