# Continuous linear operator

(Redirected from Continuous linear map)

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.

## Continuous linear operators

### Characterizations of continuity

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

1. F is continuous at 0 in X.
2. F is continuous at some point x0X.
3. F is continuous everywhere in X

and if Y is locally convex then we may add to this list:

1. for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that qFp.

and if X and Y are both Hausdorff locally convex spaces then we may add to this list:

1. F is weakly continuous and its transpose tF : Y'X' maps equicontinuous subsets of Y' to equicontinuous subsets of X'.

and if X is pseudometrizable (i.e. if it has a countable neighborhood basis at the origin) then we may add to this list:

1. F is a Bounded linear operator (i.e. it maps bounded subsets of X to bounded subsets of Y).

and if X and Y are seminormed spaces then we may add to this list:

1. for every ε > 0 there exists a δ > 0 such that ||x - y|| < δ implies ||Fx - Fy|| < ε;

and if Y is locally bounded then we may add to this list:

1. F maps some neighborhood of 0 to a bounded subset of Y.

and if X and Y are Hausdorff locally convex TVSs with Y finite-dimensional then we may add to this list:

1. the graph of F is closed in X × Y.

### Sufficient conditions for continuity

Suppose that F : XY is a linear operator between two TVSs.

• If there exists a neighborhood U of 0 in X such that F(U) is a bounded subset of Y, then F is continuous.
• If X is a pseudometrizable TVS and F maps bounded subsets of X to bounded subsets of Y, then F is continuous.

### Properties of continuous linear operators

A locally convex metrizable TVS is normable if and only if every 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

F-1(D) + x0 = F-1(D + F(x0))}}

for any subset D of Y and any x0X, which is true due to the additivity of F.

## Continuous linear functionals

Every linear functional on a 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.

### Characterizing continuous linear functionals

Let X be a topological vector space (TVS) (we do not assume that X is Hausdorff or locally convex) and let f be a linear functional on X. The following are equivalent:

1. f is continuous.
2. f is continuous at the origin.
3. f is continuous at some point of X.
4. f is uniformly continuous on X.
5. There exists some neighborhood U of the origin such that f(U) is bounded.
6. The kernel of f is closed in X.
7. Either f = 0 or else the kernel of f is not dense in X.
8. Re f is continuous, where Re f denotes the real part of f.
9. There exists a continuous seminorm p on X such that |f| ≤ p.
10. The graph of f is closed.

and if X is pseudometrizable (i.e. if it has a countable neighborhood basis at the origin) then we may add to this list:

1. f is locally bounded (i.e. it maps bounded subsets to bounded subsets).

and if in addition X is a vector space over the real numbers (which in particular, implies that f is real-valued), then we may add to this list:

1. There exists a continuous seminorm p on X such that fp.
2. For some real r, the half-space { xX : f(x) ≤ r} is closed.
3. The above statement but with the word "some" replaced by "any."

and if X is a complex topological vector space (TVS), then we may add to this list:

1. The imaginary part of f is continuous.

Thus, if X is a complex then either all three of f, Re f, and Im f are continuous (resp. bounded), or else all three are discontinuous (resp. unbounded).

### Sufficient conditions for continuous linear functionals

• Every linear function on a finite-dimensional Hausdorff topological vector space is continuous.
• If X is a TVS, then every bounded linear functional on X is continuous if and only if every bounded subset of X is contained in a finite-dimensional vector subspace.

### Properties of continuous linear functionals

If X is a complex normed space and f is a linear functional on X, then ||f|| = ||Re f|| (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 X is an open map. Note that if X is a real vector space, f is a linear functional on X, and p is a seminorm on X, then |f| ≤ p if and only if fp.