Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$.
We know $\varphi_k$ are smooth and form an orthogonal basis of $H^1_0$ and $L^2$.
Let $v \in C^\infty_c$. Define $$(-\Delta)^{\frac 12} v = \sum_{k=0}^\infty (v,\varphi_k)_{L^2}\lambda_k^{\frac 12}\varphi_k. \tag{1}$$ $(-\Delta)^{\frac 12}$ can be extended by density to the Hilbert space $$H = \{u \in L^2 \mid \lVert u \rVert_{H}^2 := \sum_{k=0}^\infty \lambda_k^{\frac 12}|(u,\varphi_k)|^2 < \infty\}.$$
Firstly, why does $v$ need to be smooth to define $(-\Delta)^{\frac 12}$ like in (1)? Something to do with the infinite sum?
Secondly, I don't understand the density statement and why $H$ is defined like it is. (1) defines $(-\Delta)^{\frac 12}:C_c^\infty \to X$ where $X$ is some space (I dunno what). $C_c^\infty$ is dense in $H^1_0$... now what?