It's evidently a subspace since if
. To prove it's closed first recall that the evaluation linear functional
is continuous since
Then, note that since
is a metric space singletons are closed and since the preimage of closed sets under continuous maps is closed it follows that
is closed. Note though that
For more general cases it can be more difficult. Showing that it's a subspace is usually simple, and showing it's closed is just usual topology/analysis.
Ask if you have more questions.