237 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.
237.1 Elementary Example
237.1.1 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.