Let $n \ge 1$ and $k \in \mathbb{N}_0$. Define the embedding map
\begin{align*}
\iota: W^{k,2}(\mathbb{R}^n) &\to H^k(\mathbb{R}^n) \\
f &\mapsto T_f,
\end{align*}
where $T_f \in \mathcal{S}'(\mathbb{R}^n)$ is the regular distribution associated to $f$, acting on test functions by $T_f(\phi) = \int_{\mathbb{R}^n} f(x)\,\phi(x)\,d\mathcal{L}^n(x)$.
Then $\iota$ is a Hilbert space isomorphism: it is linear, bijective, and the norms are equivalent. Explicitly, there exist constants $c, C > 0$ depending only on $n$ and $k$ such that
\begin{align*}
c\,\|f\|_{W^{k,2}(\mathbb{R}^n)} \le \|\iota(f)\|_{H^k(\mathbb{R}^n)} \le C\,\|f\|_{W^{k,2}(\mathbb{R}^n)}
\end{align*}
for all $f \in W^{k,2}(\mathbb{R}^n)$.
In particular:
- When $k = 0$, the map $\iota$ is an isometric isomorphism $L^2(\mathbb{R}^n) \xrightarrow{\sim} H^0(\mathbb{R}^n)$, with $\|\iota(f)\|_{H^0} = \|f\|_{L^2}$.
- For $k \ge 1$, the norms are equivalent but not equal: the equivalence constant arises from the comparison $(1 + |\xi|^2)^k \sim \sum_{|\alpha| \le k} |\xi^{2\alpha}|$, where $\sim$ denotes two-sided bounds with constants depending on $n$ and $k$.