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