Continuous linear operator

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

1. A is continuous at 0 in X.
2. A is continuous at some point x₀ in X.
3. A 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 q ∘ T ≤ p. [1]

If X and Y are seminormed spaces then we say that A is continuous if for every ${\displaystyle \epsilon >0}$ there exists a ${\displaystyle \delta >0}$ such that ${\displaystyle ||x-y||<\delta \Rightarrow ||Ax-Ay||<\epsilon }$

Suppose that A : X → Y is a linear operator between two TVSs.

• If there exists a neighborhood U of 0 in X such that A(U) is a bounded subset of Y, then A is continuous.[2]
• If X is a pseudometrizable TVS and A maps bounded subsets of X to bounded subsets of Y, then A is continuous.[2]

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]

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.[2]
6. The kernel of f is closed in X.[2]
7. Either f = 0 or else the kernel of f is not dense in X.[2]
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.

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:

10. There exists a continuous seminorm p on X such that f ≤ p.[1]

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 f ≤ p.[1] If f is a non-0 continuous linear functional then f is an open map.[1] If X is a normed space and f is a continuous linear functional on X, then ${\displaystyle \|f\|=\left\|\operatorname {Re} f\right\|}$ where Re f denotes the real part of f.[1]

Every linear function on a finite-dimensional space is continuous.

A locally convex TVS is seminormable if and only if it has a non-empty bounded open set.[2] A Hausdorff locally convex TVS is normable if and only if it has a non-empty bounded open set.

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

${\displaystyle A^{-1}(D)+x_{0}=A^{-1}(D+Ax_{0})\,\!}$

for any set D in Y and any x₀ in X, which is true due to the additivity of A.

• Conway, John B. (1990). A Course in Functional Analysis. Graduate Texts in Mathematics. 96 (2nd ed.). Springer. ISBN 0-387-97245-5.
• Dunford, Nelson (1988). Linear operators (in Romanian). New York: Interscience Publishers. ISBN 0-471-60848-3. OCLC 18412261.
• Rudin, Walter (January 1991). Functional analysis. McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• Trèves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)