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.

Properties

A continuous linear operator maps bounded sets into bounded sets. A linear functional is continuous if and only if its kernel is closed. Every linear function on a finite-dimensional space is continuous.

The following are equivalent: given a linear operator A between topological spaces X and Y:

A is continuous at 0 in X.
A is continuous at some point $x_{0}$ in X.
A is continuous everywhere in X.

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

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.

References

Rudin, Walter (January 1991). Functional Analysis. McGraw-Hill Science/Engineering/Math. ISBN 0-07-054236-8.

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert) LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition
Theorems	Banach–Alaoglu Banach–Mazur Banach–Saks Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point
Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

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