The notion of "bounded set" for a topological vector space is that of being a von Neumann bounded set.
If the space happens to also be a normed space (or a seminormed space), such as the scalar field with the absolute value for instance, then a subset is von Neumann bounded if and only if it is norm bounded; that is, if and only if
If is a set then is said to be bounded on if is a bounded subset of which if is a normed (or seminormed) space happens if and only if
A linear map is bounded on a set if and only if it is bounded on for every (because and any translation of a bounded set is again bounded).
Bounded linear maps
By definition, a linear map between TVSs is said to be bounded and is called a bounded linear operator if for every (von Neumann) bounded subset of its domain, is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain. When the domain is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if denotes this ball then is a bounded linear operator if and only if is a bounded subset of if is also a (semi)normed space then this happens if and only if the operator norm is finite. Every sequentially continuous linear operator is bounded.
Bounded on a neighborhood and local boundedness
In contrast, a map is said to be bounded on a neighborhood of a point or locally bounded at if there exists a neighborhood of this point in such that is a bounded subset of
It is "bounded on a neighborhood" (of some point) if there exists some point 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.
The term "locally bounded" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "bounded linear operator", which are related but not equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded at a point").
Bounded on a neighborhood implies continuous implies boundedEdit
For any linear map, if it is bounded on a neighborhood then it is continuous, and if it is continuous then it is bounded. The converse statements are not true in general but they are both true when the linear map's domain is a normed space. Examples and additional details are now given below.
Continuous and bounded but not bounded on a neighborhoodEdit
The next example shows that it is possible for a linear map to be continuous (and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is not always synonymous with being "bounded".
Example: A continuous and bounded linear map that is not bounded on any neighborhood: If is the identity map on some locally convex topological vector space then this linear map is always continuous (indeed, even a TVS-isomorphism) and bounded, but is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in which is equivalent to being a seminormable space (which if is Hausdorff, is the same as being a normable space).
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.
Guaranteeing that "continuous" implies "bounded on a neighborhood"
A TVS is said to be locally bounded if there exists a neighborhood that is also a bounded set. 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.
If is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood ).
Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is bounded on a neighborhood.
Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if is a TVS such that every continuous linear map (into any TVS) whose domain is is necessarily bounded on a neighborhood, then must be a locally bounded TVS (because the identity function is always a continuous linear map).
Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood.
Conversely, if is a TVS such that every continuous linear map (from any TVS) with codomain is necessarily bounded on a neighborhood, then must be a locally bounded TVS.
In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.
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.
Guaranteeing that "bounded" implies "continuous"
A continuous linear operator is always a bounded linear operator.
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.
Guaranteeing that "bounded" implies "bounded on a neighborhood"
In general, without additional information about either the linear map or its domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood".
If is a bounded linear operator from a normed space into some TVS then is necessarily continuous; this is because any open ball centered at the origin in is both a bounded subset (which implies that is bounded since is a bounded linear map) and a neighborhood of the origin in so that is thus bounded on this neighborhood of the origin, which (as mentioned above) guarantees continuity.
Continuous linear functionalsEdit
Every linear functional on a topological vector space (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them.
However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.
By definition, said to be continuous at the origin if for every open (or closed) ball of radius centered at in the codomain there exists some neighborhood of the origin in such that If is a closed ball then the condition holds if and only if
However, assuming that is instead an open ball, then is a sufficient but not necessary condition for to be true (consider for example when is the identity map on and ), whereas the non-strict inequality is instead a necessary but not sufficient condition for to be true (consider for example and the closed neighborhood ). This is one of several reasons why many definitions involving linear functionals, such as polar sets for example, involve closed (rather than open) neighborhoods and non-strict (rather than strict) inequalities.
Explicitly, this means that there exists some neighborhood of some point such that is a bounded subset of  that is, such that This supremum over the neighborhood is equal to if and only if
Importantly, a linear functional being "bounded on a neighborhood" is in general not equivalent to being a "bounded linear functional" because (as described above) it is possible for a linear map to be bounded but not continuous. However, continuity and boundedness are equivalent if the domain is a normed or seminormed space; that is, for a linear functional on a normed space, being "bounded" is equivalent to being "bounded on a neighborhood".
The equality holds for all scalars and when then will be neighborhood of the origin. So in particular, if is a positive real number then for every positive real the set is also a neighborhood of the origin and
There exists some neighborhood of the origin such that
This inequality holds if and only if for every real which shows that the positive scalar multiples of this single neighborhood will satisfy the definition of continuity at the origin given in (4) above.
By definition of the set which is called the (absolute) polar of the inequality holds if and only if Polar sets, and thus also this particular inequality, play important roles in duality theory.
Every linear map whose domain is a finite-dimensional Hausdorff topological vector space (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.
Suppose is any Hausdorff TVS. Then everylinear functional on is necessarily continuous if and only if every vector subspace of is closed. Every linear functional on is necessarily a bounded linear functional if and only if every bounded subset of is contained in a finite-dimensional vector subspace.
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