Everywhere I look in the literature, Sobolev spaces are defined on an open subset of the real line. What are the technical issues with defining a Sobolev space on a closed subset, i.e. are there problems at the boundary, and does anyone know any good references that cover this?
My main purpose is to prove H^1([0,T];\mathbb{R}) = \{ x \in L^2([0,T];\mathbb{R}) : ||x'||_{L^2} + \gamma^{2}||x||_{L^2} < \infty \} is a reproducing kernel Hilbert space. I can do this for $(0,T)$ and want to know if the proof is transferable to the case of the closed interval $[0,T]$.
Many thanks,
Matthew.