Total set

In mathematics, a total set, also sometimes called a complete set or a fundamental set, is a subset of a topological vector space whose linear span is dense in the space. In other words, a set S is total in a topological vector space V if the smallest closed subspace of V containing S is equal to V itself.

More formally, let V be a topological vector space and let S be a subset of V. The linear span of S, denoted span(S), is the set of all finite linear combinations of elements of S. The closure of span(S) is denoted cl(span(S)). The set S is a total set in V if cl(span(S)) = V.

The concept of a total set is important in functional analysis, particularly in the study of Hilbert spaces and Banach spaces. It is closely related to the concept of a basis, but a total set need not be linearly independent. For example, an orthonormal basis in a Hilbert space is a total set.

The term "complete set" can have slightly different meanings in different contexts. In some contexts, particularly in relation to orthogonal polynomials, "complete" refers to the property that no non-zero function is orthogonal to all elements of the set. This is equivalent to being a total set in the appropriate Hilbert space.