Total set

In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals [math]\displaystyle{ T }[/math] with the property that if a vector [math]\displaystyle{ x \in X }[/math] satisfies [math]\displaystyle{ f(x) = 0 }[/math] for all [math]\displaystyle{ f \in T, }[/math] then [math]\displaystyle{ x = 0 }[/math] is the zero vector. In a more general setting, a subset [math]\displaystyle{ T }[/math] of a topological vector space [math]\displaystyle{ X }[/math] is a total set or fundamental set if the linear span of [math]\displaystyle{ T }[/math] is dense in [math]\displaystyle{ X. }[/math]

See also

• Degenerate bilinear form
• Dual system
• Topologies on spaces of linear maps