is the Sobolev space $H_0^1(\Omega)$, where $\Omega$ is a bounded 1D interval, a Hilbert space?

Yes, it is Hilbert. The inner product can be taken to be $$ \langle f,g\rangle = \int_\Omega f'g', $$ or perhaps more naturally $$ \langle f,g\rangle = \int_\Omega (fg+f'g') . $$ These two inner products induce equivalent norms for $H^1_0$, as per Friedrichs' inequality.

Note however that the first one induces only a semi-norm for $H^1$.

Showing that a sequence of function does not converge uniformly [duplicate]

Rigorously defining coordinate systems on physical space

sum $\sum_{k=1}^{n-1}(1-\frac{k}{n})^{-a}\left(\frac{\log\left(k\right)}{k}-\frac{\log\left(k+1\right)}{k+1}\right)$

A question about Mayer-Vietoris sequence [duplicate]

Why is it that functions with nonisolated singularities at a point do not have Laurent series at that point?

Integral convergence $ \int_1^ \infty \frac{x+2}{(x+1)^3}dx$

Proof that if $X$ and $Y$ are independent, then $f(X)$ and $g(Y)$ are also independent

Is definable partition choice equivalent to axiom of choice over ZF?

Show, using the mean value theorem and given $|f(x)-f(y)| ≤ |x-y|^3$ and $x, y ∈ [0,1]$, that $f(x)$ has to be constant. [duplicate]

Uniform convergence of $f_n = \frac{nx}{1+nx^2}$.

Quotient of an affine scheme