In
mathematics, more specifically in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, a subset
of a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is said to be a total subset of
if the
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, ยงยง 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteriz ...
of
is a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
of
This condition arises frequently in many theorems of functional analysis.
Examples
Unbounded
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
s on
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s are defined on total subsets.
See also
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References
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{{Functional analysis
Functional analysis
Topological vector spaces