Noun
linear functional (plural linear functionals)
(mathematics) A linear mapping from a vector space to its field of scalars.
A linear functional f on a topological vector space X has either dense or closed kernel. Source: Internet
If in addition, X* separates points on X (which means that for each x ∈ X there is a linear functional in X* that's non-zero on x ) then X with this weak topology becomes Hausdorff. Source: Internet
Inverse image The set of points in the plane E that map to the same image in R under the linear functional λ define a line in E. This line is the image of the inverse map, λ −1 : R→E. Source: Internet
More generally, if V is a vector space of any dimension, then the level sets of a linear functional in V ∗ are parallel hyperplanes in V, and the action of a linear functional on a vector can be visualized in terms of these hyperplanes. Source: Internet
The set of points of a linear functional that map to zero define the kernel of the linear functional. Source: Internet
The subset S is total in X if and only if the only continuous linear functional that vanishes on S is the 0 functional: this equivalence follows from the Hahn–Banach theorem. Source: Internet