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In functional analysis and related areas of mathematics, a **continuous linear operator** or **continuous linear mapping** is a continuous linear transformation between topological vector spaces.

An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.

Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent:

- is continuous.
- is continuous at some point
- is continuous at the origin in

If is locally convex then this list may be extended to include:

- for every continuous seminorm on there exists a continuous seminorm on such that
^{[1]}

If and are both Hausdorff locally convex spaces then this list may be extended to include:

- is weakly continuous and its transpose maps equicontinuous subsets of to equicontinuous subsets of

If is a sequential space (such as a pseudometrizable space) then this list may be extended to include:

- is sequentially continuous at some (or equivalently, at every) point of its domain.

If is pseudometrizable or metrizable (such as a normed or Banach space) then we may add to this list:

- is a bounded linear operator (that is, it maps bounded subsets of to bounded subsets of ).
^{[2]}

If is seminormable space (such as a normed space) then this list may be extended to include:

- maps some neighborhood of 0 to a bounded subset of
^{[3]}

If and are both normed or seminormed spaces (with both seminorms denoted by ) then this list may be extended to include:

- for every there exists some such that

If and are Hausdorff locally convex spaces with finite-dimensional then this list may be extended to include:

- the graph of is closed in
^{[4]}

Throughout, is a linear map between topological vector spaces (TVSs).

**Bounded subset**

The notion of a "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) then a subset is von Neumann bounded if and only if it is *norm bounded*, meaning that
A subset of a normed (or seminormed) space is called *bounded* if it is norm-bounded (or equivalently, von Neumann bounded).
For example, the scalar field ( or ) with the absolute value is a normed space, so a subset is bounded if and only if is finite, which happens if and only if is contained in some open (or closed) ball centered at the origin (zero).

Any translation, scalar multiple, and subset of a bounded set is again bounded.

**Function bounded on a set**

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) if and only if it is bounded on for every non-zero scalar (because and any scalar multiple of a bounded set is again bounded).
Consequently, if is a normed or seminormed space, then a linear map 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

**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.^{[5]}

**Function 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*").

A linear map is "bounded on a neighborhood" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily continuous^{[2]} (even if its domain is not a normed space) and thus also bounded (because a continuous linear operator is always a bounded linear operator).^{[6]}

For any linear map, if it is bounded on a neighborhood then it is continuous,^{[2]}^{[7]} and if it is continuous then it is bounded.^{[6]} 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.

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

To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being bounded, and being bounded on a neighborhood are all equivalent.
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.^{[6]}

**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.^{[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.
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.^{[8]}
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.^{[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.

**Guaranteeing that "bounded" implies "continuous"**

A continuous linear operator is always a bounded linear operator.^{[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.^{[2]}
The same is true of a linear map from a bornological space into a locally convex space.^{[6]}

**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.

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.

Let be a topological vector space (TVS) over the field ( need not be Hausdorff or locally convex) and let be a linear functional on
The following are equivalent:^{[1]}

- is continuous.
- is uniformly continuous on
- is continuous at some point of
- is continuous at the origin.
- 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
- It is important that be a closed ball in this supremum characterization. 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.

- It is important that be a closed ball in this supremum characterization. Assuming that is instead an open ball, then is a sufficient but

- is bounded on a neighborhood (of some point). Said differently, is a locally bounded at some point of its domain.
- Explicitly, this means that there exists some neighborhood of some point such that is a bounded subset of
^{[2]}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".

- Explicitly, this means that there exists some neighborhood of some point such that is a bounded subset of
- is bounded on a neighborhood of the origin. Said differently, is a locally bounded at the origin.
- 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 a neighborhood of the origin and Using proves the next statement when

- 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 so also this particular inequality, play important roles in duality theory.

- is a locally bounded at every point of its domain.
- The kernel of is closed in
^{[2]} - Either or else the kernel of is
*not*dense in^{[2]} - There exists a continuous seminorm on such that
- In particular, is continuous if and only if the seminorm is a continuous.

- The graph of is closed.
^{[9]} - is continuous, where denotes the real part of

If and are complex vector spaces then this list may be extended to include:

- The imaginary part of is continuous.

If the domain is a sequential space then this list may be extended to include:

- is sequentially continuous at some (or equivalently, at every) point of its domain.
^{[2]}

If the domain is metrizable or pseudometrizable (for example, a Fréchet space or a normed space) then this list may be extended to include:

- is a bounded linear operator (that is, it maps bounded subsets of its domain to bounded subsets of its codomain).
^{[2]}

If the domain is a bornological space (for example, a pseudometrizable TVS) and is locally convex then this list may be extended to include:

- is a bounded linear operator.
^{[2]} - is sequentially continuous at some (or equivalently, at every) point of its domain.
^{[10]} - is sequentially continuous at the origin.

and if in addition is a vector space over the real numbers (which in particular, implies that is real-valued) then this list may be extended to include:

- There exists a continuous seminorm on such that
^{[1]} - For some real the half-space is closed.
- For any real the half-space is closed.
^{[11]}

If is complex then either all three of and are continuous (respectively, bounded), or else all three are discontinuous (respectively, unbounded).

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.

Every (constant) map between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood of the origin. In particular, every TVS has a non-empty continuous dual space (although it is possible for the constant zero map to be its only continuous linear functional).

Suppose is any Hausdorff TVS. Then *every* linear functional on is necessarily continuous if and only if every vector subspace of is closed.^{[12]} 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.^{[13]}

A locally convex metrizable topological vector space is normable if and only if every bounded linear functional on it is continuous.

A continuous linear operator maps bounded sets into bounded sets.

The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality for any subset of and any which is true due to the additivity of

If is a complex normed space and is a linear functional on then ^{[14]} (where in particular, one side is infinite if and only if the other side is infinite).

Every non-trivial continuous linear functional on a TVS is an open map.^{[1]}
If is a linear functional on a real vector space and if is a seminorm on then if and only if ^{[1]}

If is a linear functional and is a non-empty subset, then by defining the sets the supremum can be written more succinctly as because If is a scalar then so that if is a real number and is the closed ball of radius centered at the origin then the following are equivalent:

- Bounded linear operator – Linear transformation between topological vector spaces
- Compact operator – Type of continuous linear operator
- Continuous linear extension – Mathematical method in functional analysis
- Contraction (operator theory) – Bounded operators with sub-unit norm
- Discontinuous linear map
- Finest locally convex topology – A vector space with a topology defined by convex open sets
- Linear functionals – Linear map from a vector space to its field of scalars
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Positive linear functional – ordered vector space with a partial order
- Topologies on spaces of linear maps
- Unbounded operator – Linear operator defined on a dense linear subspace

- ^
^{a}^{b}^{c}^{d}^{e}Narici & Beckenstein 2011, pp. 126–128. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}Narici & Beckenstein 2011, pp. 156–175. **^**Wilansky 2013, p. 54.**^**Narici & Beckenstein 2011, p. 476.**^**Wilansky 2013, pp. 47–50.- ^
^{a}^{b}^{c}^{d}^{e}Narici & Beckenstein 2011, pp. 441–457. **^**Wilansky 2013, pp. 54–55.- ^
^{a}^{b}^{c}^{d}Wilansky 2013, pp. 53–55. **^**Wilansky 2013, p. 63.**^**Narici & Beckenstein 2011, pp. 451–457.**^**Narici & Beckenstein 2011, pp. 225–273.**^**Wilansky 2013, p. 55.**^**Wilansky 2013, p. 50.**^**Narici & Beckenstein 2011, p. 128.

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