# Proving the density of a set into another set

I am dealing with the proofs for some density theorems in the context of Functional Analysis. I know that one can prove that $\bar{X} = Y$ by means of constructing a sequence $\left\{f_n \right\}_n \subset X$ such that $\|f_n-f\|_Y \to 0$ for each $f \in Y$. However, sometimes finding such sequences can be very complicated.

My question is if the following approach is also correct, as I don't seem to get along very well with the theory: can I prove that $X \subset Y$ and then that the inclusion map is continuous via, for example, the Closed Graph Theorem?

I am restricting to the functional analysis background, so let's say that $X$ and $Y$ are Fréchet spaces.

Any light you may throw to the topic would be greatly appreciated.

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If $X$ is a topological subspace of $Y$, isn't the inclusion map continuous for granted? –  Ilya Jan 31 at 17:59