Below, $\mathcal{C} = \Omega \times (0,\infty)$, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$, and $\Omega$ is a bounded smooth domain. $tr_\Omega:H^1(C) \to L^2(\Omega)$ refers to the trace operator $tr_\Omega u = u(\cdot,0)$. 
How do I know that the constant functions are in that bigger space (let's just take $\epsilon =1$)? They obviously have finite $H^\epsilon(\mathcal{C})$ norm but surely that is not enough?
Old version of the question: Let $X$ be a Hilbert space with norm $|\cdot|_X$. Define $Y$ as the completion of $X$ under a function (norm?) $|\cdot|_Y$, which satisfies $|x|_Y \leq C|x|_X$ when $x \in X$.
I guess then $y \in Y$ means that there exist $x_n \in X$ such that $(x_n)$ converges to $y$ in the $Y$ norm.
Is it possible to say that if $|y|_Y < \infty$, then $y \in Y$? That is, if $y$ has a finite norm (think of $Y$ as some Sobolev space and $y$ here as some function) then is it in $Y$?
I'm not sure your question is well posed. When you say "if $|y|_Y < \infty $" then it seems you are implicitly assuming that $y$ is an element of $Y$. Indeed if $y$ were not an element of $Y$, then the expression $|y|_Y < \infty $ would not be defined.