Continuous linear operator


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 is a linear operator between two topological vector spaces.
The following are equivalent:

  1. is continuous at 0 in.
  2. is continuous at some point.
  3. is continuous everywhere in
and if is locally convex then we may add to this list:

  1. for every continuous seminorm on, there exists a continuous seminorm on such that.
and if is locally bounded then we may add to this list:

  1. maps some neighborhood of 0 to a bounded subset of.
and if and are both Hausdorff locally convex spaces then we may add to this list:

  1. is weakly continuous and its transpose maps equicontinuous subsets of to equicontinuous subsets of.
and if and are seminormed spaces then we may add to this list:

  1. for every there exists a such that
    implies ;
and if and are Hausdorff locally convex TVSs with finite-dimensional then we may add to this list:

  1. the graph of is closed in.

Sufficient conditions for continuity

Suppose that is a linear operator between two TVSs.

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

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 be a topological vector space and let be a linear functional on.
The following are equivalent:

  1. is continuous.
  2. is continuous at the origin.
  3. is continuous at some point of.
  4. is uniformly continuous on.
  5. There exists some neighborhood of the origin such that is bounded.
  6. The kernel of is closed in.
  7. Either or else the kernel of is not dense in.
  8. is continuous, where denotes the real part of.
  9. There exists a continuous seminorm on such that.
  10. The graph of is closed.
and if in addition is a vector space over the real numbers, then we may add to this list:

  1. There exists a continuous seminorm on such that.
  2. For some real, the half-space is closed.
  3. The above statement but with the word "some" replaced by "any."
and if is a complex topological vector space, then we may add to this list:

  1. is continuous.
Thus, if is a complex then either all three of,, and are continuous, or else all three are discontinuous.

Sufficient conditions for continuous linear functionals

Properties of continuous linear functionals

If is a complex normed space and is a linear functional on, then .
Every non-trivial continuous linear functional on a TVS is an open map.
Note that if is a real vector space, is a linear functional on, and is a seminorm on, then if and only if.