Rellich Embedding Theorem
Let $\Omega$ be a bounded domain, then $H_{0}^{1}(\Omega)$ is compactly embedded in $L^{p}(\Omega)$ (denoted by $H_{0}^{1}(\Omega)\subset\subset L^{p}(\Omega)$) for $1\leq p <2^{*}$.
I apologize beforehand to ask "stupid" question but precisely speaking, I should have
\begin{align*}
\exists T:H_{0}^{1}(\Omega)\to L^{p}(\Omega)\text{ injective mapping }s.t.\\
\end{align*}
(i) $\exists C>0\forall u\in H_{0}^{1}(\Omega)\,s.t.\,||Tu||_{L^{p}(\Omega)}\leq ||u||_{H_{0}^{1}(\Omega)}$
(ii) $\forall (u_{n})_{n\in\mathbb{N}}\subset H_{0}^{1}(\Omega)$ bounded in $H_{0}^{1}(\Omega)$, $\exists(Tu_{n_{k}})_{k\in\mathbb{N}} \subset (Tu_{n})_{n\in\mathbb{N}}$ such that $||Tu_{n_{k}}-Tu||_{L^{p}}\to 0$ as $k\to\infty$.
So, my question is although we have $Tu$ but why do we often denote it simply by $u$?
Can we really say $||u_{n_{k}}-u||_{L^{p}(\Omega}\to 0$ as $k\to\infty$ and $||Tu||_{L^{p}(\Omega)}=||u||_{L^{p}(\Omega)}$? I want to clarify this part in a very rigorous way.
Thank you for any help!