[proofplan] The Helmholtz operator $1 - \Delta$ acts on the Fourier side as multiplication by $(1 + |\xi|^2)$, which is strictly positive and finite for every $\xi$. We show the inverse map $(1+|\xi|^2)^{-1}$ defines an isometry $H^s \to H^{s+2}$, the forward map defines an isometry $H^{s+2} \to H^s$, and the two are mutual inverses pointwise in Fourier space. [/proofplan] [step:Define the inverse operator in Fourier space] Let $f \in H^s(\mathbb{R}^n)$ and define $u = (1-\Delta)^{-1}f$ by the Fourier multiplier formula: \begin{align*} \hat{u}(\xi) &:= \frac{\hat{f}(\xi)}{1+|\xi|^2} \quad \text{for every } \xi \in \mathbb{R}^n. \end{align*} This is well-defined since $1 + |\xi|^2 \ge 1 > 0$ for all $\xi \in \mathbb{R}^n$. [/step] [step:Verify that $(1-\Delta)^{-1}: H^s \to H^{s+2}$ is an isometry] Compute the $H^{s+2}$ norm of $u$ directly: \begin{align*} \|u\|_{H^{s+2}}^2 &= \int_{\mathbb{R}^n}(1+|\xi|^2)^{s+2}|\hat{u}(\xi)|^2\,d\mathcal{L}^n(\xi) \\ &= \int_{\mathbb{R}^n}(1+|\xi|^2)^{s+2}\frac{|\hat{f}(\xi)|^2}{(1+|\xi|^2)^2}\,d\mathcal{L}^n(\xi) \\ &= \int_{\mathbb{R}^n}(1+|\xi|^2)^{s}|\hat{f}(\xi)|^2\,d\mathcal{L}^n(\xi) \\ &= \|f\|_{H^s}^2. \end{align*} So $u \in H^{s+2}(\mathbb{R}^n)$ and $\|u\|_{H^{s+2}} = \|f\|_{H^s}$. [/step] [step:Verify that $(1-\Delta): H^{s+2} \to H^s$ is an isometry] For $u \in H^{s+2}(\mathbb{R}^n)$, set $f = (1-\Delta)u$, so $\hat{f}(\xi) = (1+|\xi|^2)\hat{u}(\xi)$. Then: \begin{align*} \|f\|_{H^s}^2 &= \int_{\mathbb{R}^n}(1+|\xi|^2)^s|\hat{f}(\xi)|^2\,d\mathcal{L}^n(\xi) \\ &= \int_{\mathbb{R}^n}(1+|\xi|^2)^s(1+|\xi|^2)^2|\hat{u}(\xi)|^2\,d\mathcal{L}^n(\xi) \\ &= \|u\|_{H^{s+2}}^2. \end{align*} [/step] [step:Establish bijectivity by showing the maps are mutual inverses] The maps $(1-\Delta): H^{s+2} \to H^s$ and $(1-\Delta)^{-1}: H^s \to H^{s+2}$ are mutual inverses, since $(1+|\xi|^2)\cdot(1+|\xi|^2)^{-1} = 1$ pointwise in Fourier space for every $\xi \in \mathbb{R}^n$. Both are bounded [linear maps](/page/Linear%20Map), and each is an isometry by the preceding steps, so both are isometric isomorphisms. [/step]