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In the framework of quasi-regular strongly local Dirichlet form$(\mathscr{E},D(\mathscr{E}))$ on $L^2(X;\mathfrak{m})$ admitting minimal$\mathscr{E}$-dominant measure $\mu$, we construct a natural $p$-energyfunctional $(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$ on $L^p(X;\mathfrak{m})$and $(1,p)$-Sobolev space $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ for$p\in]1,+\infty[$. In this paper, we establish the Clarkson type inequality for$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$. As a consequence,$(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$ is a uniformly convex Banach space, hence itis reflexive. Based on the reflexivity of $(H^{1,p}(X),\|\cdot\|_{H^{1,p}})$,we prove that (generalized) normal contraction operates on$(\mathscr{E}^{\,p},D(\mathscr{E}^{\,p}))$, which has been shown in the case ofvarious concrete settings, but has not been proved for such general framework.Moreover, we prove that $(1,p)$-capacity ${\rm Cap}_{1,p}(A)<\infty$ for openset $A$ admits an equilibrium potential $e_A\in D(\mathscr{E}^{\,p})$ with$0\leq e_A\leq 1$ $\mathfrak{m}$-a.e. and $e_A=1$ $\mathfrak{m})$-a.e.~on $A$.

(1,p)-Sobolev spaces based on strongly local Dirichlet forms