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.