is a linear operator between two topological vector spaces (TVSs).
are both Hausdorff locally convex spaces then this list may be extended to include: If
is pseudometrizable or metrizable (such as a normed or Banach space) then we may add to this list: If
is a linear map between topological vector spaces (TVSs).
is contained in some open (or closed) ball centered at the origin (zero).
is bounded on some (equivalently, on every) non-degenerate open or closed ball (not necessarily centered at the origin, and of any radius) if and only if it is bounded on the closed unit ball centered at the origin
is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin.
[5] Function bounded on a neighborhood and local boundedness In contrast, a map
in its domain at which it is locally bounded, in which case this linear map
is necessarily locally bounded at every point of its domain.
This shows that it is possible for a linear map to be continuous but not bounded on any neighborhood.
Indeed, this example shows that every locally convex space that is not seminormable has a linear TVS-automorphism that is not bounded on any neighborhood of any point.
Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
A linear map whose domain or codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood.
And a bounded linear operator valued in a locally convex space will be continuous if its domain is (pseudo)metrizable[2] or bornological.
[8] For example, every normed or seminormed space is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin.
must be a locally bounded TVS (because the identity function
[8] In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.
[8] Thus when the domain or the codomain of a linear map is normable or seminormable, then continuity will be equivalent to being bounded on a neighborhood.
[6] But importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be bounded but to not be continuous.
A linear map whose domain is pseudometrizable (such as any normed space) is bounded if and only if it is continuous.
be a topological vector space (TVS) over the field
are complex vector spaces then this list may be extended to include: If the domain
is a sequential space then this list may be extended to include: If the domain
is locally convex then this list may be extended to include: and if in addition
is a vector space over the real numbers (which in particular, implies that
Every linear map whose domain is a finite-dimensional Hausdorff topological vector space (TVS) is continuous.
[13] A locally convex metrizable topological vector space is normable if and only if every bounded linear functional on it is continuous.
Every non-trivial continuous linear functional on a TVS
is a linear functional on a real vector space