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 :

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

Then, if , one has the imbedding , and

Comments

For references cf. Imbedding theorems.