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Mathematics for Physics

361 Sets

A collection of mathematical objects such as numbers or vectors that is used as the basic building block for other constructions.

definition (Complete Set) A subset of a normed space whose span is dense in that space, meaning any element in the space can be represented by a linear combination of the set’s members. The set must satisfy the following condition:

  • The only vector in the space that is orthogonal to every element in the set is the zero vector.

where

  • \(M\) is the set of vectors being tested for completeness.
  • \(H\) is the Hilbert space containing the set.
  • \(\text{span } M\) is the set of all finite linear combinations of elements in \(M\).
  • \(\delta(x - x')\) is the Dirac delta function.
  • \(1 = \sum |e_n\rangle\langle e_n|\) is the completeness relation for a discrete orthonormal basis.

Note:

  • the same notion applies in an inner product space.
  • \(M\) may be a set of functions.
  • an element may be approximated rather than exactly represented by a linear combination.
  • \(\delta(x - x')\) is used to express the completeness relation for sets with continuous indices.
  • the completeness relation is also called the resolution of the identity.

361.1 Examples

361.1.1 Simple

A set is a collection of elements. Here a three-element set of numbers.

\[ A = \{ 1,\ 2,\ 3 \} \]

where

  • \(A\) is the set.
  • \(1, 2, 3\) are its elements.

361.1.2 General

Sets of vectors are the building blocks of linear spaces. Here a basis set in \(\mathbb{R}^{3}\).

\[ B = \{ e_{1},\ e_{2},\ e_{3} \} \]

\[ e_{1} = (1,0,0),\quad e_{2} = (0,1,0),\quad e_{3} = (0,0,1) \]

where

  • \(B\) is a set of vectors.