Continuous operator
A continuous mapping 
of a subset 
of a topological space 
, which as a role is also a vector space, into a space 
of the same type, specifically: A mapping 
, 
, is continuous at a point 
if for any neighbourhood 
of the point 
there is a neighbourhood 
of 
such that 
; a mapping 
is continuous on a set 
if it is continuous at every point of 
.
For an operator 
to be continuous on 
it is necessary and sufficient that for every open (closed) set 
the complete inverse image 
is the trace on 
of an open (closed) set in 
, that is, 
, where 
is open (closed) in 
. For continuous operators the chain rule holds: Suppose that 
, 
, is continuous on 
(or at 
) and suppose also that 
, 
, is continuous on 
(or at 
). If 
is not empty (or 
), then 
is continuous on 
(or at 
).
When 
and 
are topological vector spaces and 
is a linear continuous operator on a linear subspace 
with values in 
, then the continuity of 
at some point of 
, say, at the origin, implies the continuity of 
on the whole of 
. A continuous operator on a submanifold 
of a topological vector space 
is bounded on 
, that is, the image of any bounded set 
is bounded in 
. If 
and 
are separable, then the compactness of 
implies that of 
.
An operator 
is uniformly continuous on 
if for any neighbourhood of the origin 
there exists a neighbourhood of the origin 
such that 
implies 
. An operator that is linear and continuous on a linear submanifold of a topological vector space is automatically uniformly continuous on this submanifold.
Apart from continuity one introduces the concept of countable continuity of an operator. An operator 
is countably continuous at 
if 
for any sequence 
, 
. For metrizable spaces continuity and countable continuity coincide.
References
| [1] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |
| [2] | L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1974) (Translated from Russian) |
Comments
In Western literature the term "operatoroperator" tends to be reserved for a mapping between vector spaces. See [a1], [a2].
References
| [a1] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |
| [a2] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |
Continuous operator. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuous_operator&oldid=14154