In my work, I encountered the following problem.
Consider the set of real-valued functions, which are ``balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \infty} f(x)=\pm\lambda$ for some $\lambda\in \mathbb{R}$. That is, bounded functions whose limits at $x=\pm \infty$ exist and are additive opposite. I would like to make this a Hilbert space. The inner product cannot involve the function themselves but, if I assume the existence of first derivatives (in some sense), I would like to define my inner product to be $$(f,g)=\int_{-\infty}^\infty f'g' dx.$$ Is there a way to restrict the functions (infinitely differentiable for example) so that we have a Hilbert space?
If you want to solve a related but simpler problem and forego the balance part, consider functions in $H^1(\mathbb{R})$. Is there a way to restrict the functions (that is consider a subset of $H^1$) such that this subset of $H^1$ endowed with the inner product above is a Hilbert space.
In a nutshell, because my inner product cannot involve the functions themselves (because the functions do not converge to zero at $\infty$), I would like my inner product to involve the derivatives only.