As per wiki here https://en.wikipedia.org/wiki/Continuous_linear_extension the concept continuous extension is explained as: we can extend a linear transformation domain to a larger space if the smaller space is dense in it. Is density enough to do that? Coz I read somewhere you need the smaller space to be closed as well. Could anyone give clear definition to "extension by continuity" and examples as well? in which particular cases "extension by continuity" is important?
Thanks in advance.
Edit: I tagged the wiki page and hahn-banach-theorem, anything tangential to that is the considered context. Elsewise I am happy to provide more context.