Total set

This article is about a concept in functional analysis. For the type of metric space, see Complete space.

In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals ${\displaystyle T}$ with the property that if a vector ${\displaystyle x\in X}$ satisfies ${\displaystyle f(x)=0}$ for all ${\displaystyle f\in T,}$ then ${\displaystyle x=0}$ is the zero vector.^[1]

In a more general setting, a subset ${\displaystyle T}$ of a topological vector space ${\displaystyle X}$ is a total set or fundamental set if the linear span of ${\displaystyle T}$ is dense in ${\displaystyle X.}$^[2]

See also

• Kadec norm – All infinite-dimensional, separable Banach spaces are homeomorphicPages displaying short descriptions of redirect targets
• Degenerate bilinear form
• Dual system
• Topologies on spaces of linear maps