I am reading a paper that defines a Gelfand triples. The paper states:
"We define the Gelfand triple of Hilbert spaces $V \subset H \subset V^*$ by
$$H= (L^2(D), <\cdot, \cdot>, ||\cdot||), \quad V= (H_0^1 (D), <\nabla\cdot, \nabla\cdot>, ||\cdot||_V = || \nabla\cdot ||)$$
where $D \subset \mathbb{R}^d$ denotes a bounded open set, with Lipschitz boundary $\partial D$, and $v^*$ is the dual of $V$ with respect to the pairing induced by $H$."
What is $L^2(D)$ and what is $H_0^1(D)$? How would I search for this (on google or here)?
I would guess $L^2(D)$ is functions that are $L^2$ integrable on D, i.e. $\int_D |f|^2 d\mu < \infty$ (though I'm not sure what the measure $\mu$ is in this context). What about $H_0^1$?