Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I often see something like "$V \subset H$ is continuous embedding", meaning that the inclusion map $i:V \to H$ defined $i(v) = v \in H$ for $v \in V$ is continuous: so $$|i(v)|_H \leq C|v|_V$$ holds for some constant $C$.

Why is this so important?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

I assume you are talking about Banach spaces here.

Continuous embeddings are important because they preserve the structure of the smaller Banach space and allow us to regard it as a subspace of the larger.

Convergence of the sequence $(x_n)$ in the space $V$ implies convergence of $(i(x_n))$ in the space $i(V)$ for instance.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.