Imbedding of function spaces

A set-theoretic inclusion $ V \subset W $ of a linear normed space $ V $ into a linear normed space $ W $, for which the following inequality is valid for any $ x \in V $:

$$ \| x \| _ {W} \leq C \| x \| _ {V} , $$

where $ C $ is a constant which does not depend on $ x \in V $. Here, $ \| x \| _ {W} $ is the norm (semi-norm) of the element $ x $ in $ W $, while $ \| x \| _ {V} $ is the norm (semi-norm) of the element $ x $ in $ V $.

The identity operator from $ V $ into $ W $, which assigns to an element $ x \in V $ the same element seen as an element of $ W $, is said to be the imbedding operator of $ V $ into $ W $. 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 $ E $ be a Lebesgue-measurable set in the $ n $- dimensional Euclidean space with finite measure $ \mathop{\rm mes} E $ and let $ L _ {p} ( E) $, $ 1 \leq p \leq \infty $, be the Lebesgue space of measurable functions which are $ p $- th power summable over $ E $ with norm

$$ \| x \| _ {p} = \ \left [ \int\limits _ { E } | x ( t) | ^ {p} dt \right ] ^ {1/p} . $$

Then, if $ p \geq q $, one has the imbedding $ L _ {p} ( E) \rightarrow L _ {q} ( E) $, and

$$ \| x \| _ {q} \leq \ ( \mathop{\rm mes} E) ^ {1/q- 1/p } \| x \| _ {p} . $$

Comments

For references cf. Imbedding theorems.