220 Complete Set
A set of vectors whose linear combinations can approximate every vector in a larger set that is used as a basis-like tool to represent other vectors.
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.
220.1 Elementary Example
220.1.1 Simple
A complete set spans a dense subspace. In \(\mathbb{R}^{2}\), the two standard basis vectors form a complete set.
\[ M = \{ e_{1},\ e_{2} \} \]
\[ e_{1} = (1,0),\quad e_{2} = (0,1) \]
\[ \operatorname{span} M = \mathbb{R}^{2} \]
where
- \(M\) is the set being tested for completeness.
- \(\operatorname{span} M\) is the set of finite linear combinations of members of \(M\).
220.1.2 General
In \(\mathbb{R}^{3}\), the three standard basis vectors form a complete set: only the zero vector is orthogonal to all three.
\[ M = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ \operatorname{span} M = \mathbb{R}^{3} \]
where
- if \(\langle v, e_{i} \rangle = 0\) for \(i = 1,2,3\), then \(v = 0\).