Imbedding of function spaces
A set-theoretic inclusion of a linear normed space
into a linear normed space
, for which the following inequality is valid for any
:
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where is a constant which does not depend on
. Here,
is the norm (semi-norm) of the element
in
, while
is the norm (semi-norm) of the element
in
.
The identity operator from into
, which assigns to an element
the same element seen as an element of
, is said to be the imbedding operator of
into
. The imbedding operator is always bounded. If the imbedding operator is a completely-continuous operator, the imbedding of function spaces is said to be compact. Facts on imbedding of function spaces are established by so-called imbedding theorems.
Example. Let be a Lebesgue-measurable set in the
-dimensional Euclidean space with finite measure
and let
,
, be the Lebesgue space of measurable functions which are
-th power summable over
with norm
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Then, if , one has the imbedding
, and
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Comments
For references cf. Imbedding theorems.
Imbedding of function spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_of_function_spaces&oldid=11516