Hilbert Spaces: Infinite-Dimensional Vector Spaces

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1 Hilbert Spaces: Infinite-Dimensional Vector Spaces PHYS Southern Illinois University October 27, 2016 PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

2 Infinite dimensional vector spaces are vector spaces that cannot be spanned by a finite number of elements. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

3 Infinite dimensional vector spaces are vector spaces that cannot be spanned by a finite number of elements. Example (l 2 ) A prime example of an infinite-dimensional vector space is l 2. This is the subset of infinite-length sequences: { } l 2 := x = (x 1, x 2, ) C : x k 2 <. k=1 PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

4 Infinite dimensional vector spaces are vector spaces that cannot be spanned by a finite number of elements. Example (l 2 ) A prime example of an infinite-dimensional vector space is l 2. This is the subset of infinite-length sequences: { } l 2 := x = (x 1, x 2, ) C : x k 2 <. k=1 Vector addition in l 2 is defined component-wise: x + y = (x 1, x 2, ) + (y 1, y 2, ) := (x 1 + y 1, x 2 + y 2, ). PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

5 Properties of l 2 l 2 has an inner product defined as (x, y) = xk y k. k=1 The norm of a vector x l 2 is given by X = (x, x). PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

6 Properties of l 2 l 2 has an inner product defined as (x, y) = xk y k. k=1 The norm of a vector x l 2 is given by X = (x, x). Note that (x, y) is finite for x, y l 2 since xk y k 1 ( xk 2 + y k 2) <. 2 k=1 k=1 PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

7 Properties of l 2 l 2 is a separable vector space. Being separable means that it has a countable basis. The basis for l 2 is given by e 1 = (1, 0, 0, ), e 2 = (0, 1, 0, ),, e n = (0,, 1, 0, ), PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

8 Properties of l 2 l 2 is a separable vector space. Being separable means that it has a countable basis. The basis for l 2 is given by e 1 = (1, 0, 0, ), e 2 = (0, 1, 0, ),, e n = (0,, 1, 0, ), Properties of l 2 l 2 is a complete vector space. This means that every Cauchy sequence defined in l 2 converges in l 2. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

9 Properties of l 2 l 2 is a separable vector space. Being separable means that it has a countable basis. The basis for l 2 is given by e 1 = (1, 0, 0, ), e 2 = (0, 1, 0, ),, e n = (0,, 1, 0, ), Properties of l 2 l 2 is a complete vector space. This means that every Cauchy sequence defined in l 2 converges in l 2. Proof. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

10 Hilbert Space A Hilbert space is an infinite-dimensional inner product space that is both separable and complete. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

11 Hilbert Space A Hilbert space is an infinite-dimensional inner product space that is both separable and complete. Let H be a Hilbert space. A set of vectors {φ 1, φ 2, } with φ k H is said to be an orthonormal system if (φ i, φ j ) = δ ij = 0. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

12 Hilbert Space A Hilbert space is an infinite-dimensional inner product space that is both separable and complete. Let H be a Hilbert space. A set of vectors {φ 1, φ 2, } with φ k H is said to be an orthonormal system if (φ i, φ j ) = δ ij = 0. An orthonormal system {φ 1, φ 2, } is said to be complete if and only if the only vector orthogonal to each of the φ k is the all zero vector 0. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

13 Hilbert Space A Hilbert space is an infinite-dimensional inner product space that is both separable and complete. Let H be a Hilbert space. A set of vectors {φ 1, φ 2, } with φ k H is said to be an orthonormal system if (φ i, φ j ) = δ ij = 0. An orthonormal system {φ 1, φ 2, } is said to be complete if and only if the only vector orthogonal to each of the φ k is the all zero vector 0. Note An orthonormal set of vectors {φ 1, φ 2, } being complete is different than a vector space being complete. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

14 Complete sets of vectors Theorem Let {φ 1, φ 2, } be an orthonormal set for a Hilbert space H. The following statements are equivalent: 1 The set {φ 1, φ 2, } is complete. 2 Every vector x H can be expressed as x = k=1 (φ k, x)φ k. 3 Every vector x H satisfies x 2 = k=1 (φ k, x) 2. 4 Every pair of vectors x, y H satisfies (x, y) = k=1 (x, φ k)(φ k, y). PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6

15 Complete sets of vectors Theorem Let {φ 1, φ 2, } be an orthonormal set for a Hilbert space H. The following statements are equivalent: 1 The set {φ 1, φ 2, } is complete. 2 Every vector x H can be expressed as x = k=1 (φ k, x)φ k. 3 Every vector x H satisfies x 2 = k=1 (φ k, x) 2. 4 Every pair of vectors x, y H satisfies (x, y) = k=1 (x, φ k)(φ k, y). Proof. PHYS Southern Illinois University Hilbert Spaces: Infinite-Dimensional Vector Spaces October 27, / 6