I am trying to show that the elliptic differential operator on the weighted $H^1(M)$ space ($M \subset \mathbb{R}^{n+1}$ is a compact Riemannian hypersurface)
$$\begin{align*} L: H_w^1(M) &\longrightarrow (H_w^1(M))'\\ u &\mapsto Lu := \Delta u + |A|^2 u + \frac{1}{2} u - \frac{1}{2} \langle x, \nabla u \rangle. \end{align*}$$
of this paper is continuous, which is defined on page $26$ (it is the page $780$ with respect to the numeration of the journal). I would like to know if my attempt is correct.
$\textbf{My attempt:}$
The weighted $H^1(M)$ space is endowed with the inner product $\langle u, v \rangle_w := \int_M uv e^{-\frac{|x|^2}{4}}$.
Define
$$\begin{align*} \varphi_v: H_w^1(M) &\longrightarrow \mathbb{R}\\ u &\mapsto \varphi_v(u) := \langle u, v \rangle_w. \end{align*}$$
for each $v \in H_w^1(M)$ fixed.
By Riesz Representation Theorem, $H_w^1(M)$ and its dual are isomorphic so that $\langle v, Lu \rangle_w$ makes sense less isomorphism, then $\langle v, Lu \rangle_w \equiv \varphi_v \circ L$. Now, $\varphi_v \circ L$ is continuous in the strong topology because $\langle \cdot, \cdot \rangle_w$ is continuous in the strong topology, therefore $\varphi_v \circ L$ is continuous in the weak topology by
$\textbf{Theorem 3.10.}$ Let $E$ and $F$ be two Banach spaces and let $T$ be a linear operator from $E$ into $F$. Assume that $T$ is continuous in the strong topologies. Then $T$ is continuous from $E$ weak $\sigma(E, E^{\star})$ into $F$ weak $\sigma(F, F^{\star})$ and conversely.
on page $61$ of the book "Functional Analysis, Sobolev spaces and Partial Differential Equations" by Haim Brezis.
and $L$ is continuous in the weak topology by
$\textbf{Proposition 3.2.}$ Let $Z$ be a topological space and let $\psi$ be a map from $Z$ into $X$. Then $\psi$ is continuous if and only if $\varphi_i \circ \psi$ is continuous from $Z$ into $Y_i$ for every $i \in I$.
on page $56$ of the book "Functional Analysis, Sobolev spaces and Partial Differential Equations" by Haim Brezis.
Thus, $L$ is continuous in the strong topology by the theorem $3.10$ above. $\square$
$\textbf{P.S.:}$ proposition $3.2$ considers continuity for the spaces endowed with weak topologies according Brezis.
Thanks in advance!
$\textbf{EDIT:}$ $\{ \varphi_i \}_{i \in I}$ such that for every $i \in I$ of the proposition $3.2$ is a collection such that $\varphi_i$ maps $X$ into $Y_i$, where $X$ is assumed a set without any structure and $\{ Y_i \}_{i \in I}$ is a collection of topological spaces (it is on the first paragraph of the page $55$ of the Brezis' book).