The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with $L^p_{\text{loc}}(\mathbb{R})$ we look at those elements finite under the seminorm

$$ \|f\|_{M^p} = \lim_{R\to\infty} \left(\frac1{2R}\int_{-R}^R \left|f(x)\right|^pdx\right)^{1/p} $$

Modding out by the zero elements nets us a Banach space for $1 \le p <\infty$, and a Hilbert space with the natural inner product for $p=2$. We'll call this space $M^p$.

We can see that $M^2$ is an example of a nonseparable Hilbert space because the collection $e^{i\xi x}$ is orthonormal for all $\xi \in \mathbb{R}$. We can look at the subspace $B^p\subseteq M^p$ of elements spanned by these functions, called the Besicovitch almost periodic functions.

We can see that $B^2\neq M^2$ since there are functions like

$$ f(x) = \left\{\begin{align}1 \ \ \ \ \ x\ge0 \\ -1 \ \ \ \ \ x < 0\end{align}\right. $$

which is orthogonal to all $e^{i\xi x}$, and $\|f\|_2 = 1$.

Question: I can't seem to find any discussion of $M^p$ independent from $B^p$. Is there a standard name for $M^p$? Is there a convenient description of an orthonormal basis for $M^2$?

The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with $L^p_{\text{loc}}(\mathbb{R})$ we look at those elements finite under the seminorm

$$ \|f\|_{M^p} = \limsup_{R\to\infty} \left(\frac1{2R}\int_{-R}^R \left|f(x)\right|^pdx\right)^{1/p} $$

Modding out (and completing?) by the zero elements nets us a Banach space for $1 \le p <\infty$, and a Hilbert space with the natural inner product for $p=2$. We'll call this space $M^p$.

We can see that $M^2$ is an example of a nonseparable Hilbert space because the collection $e^{i\xi x}$ is orthonormal for all $\xi \in \mathbb{R}$. We can look at the subspace $B^p\subseteq M^p$ of elements spanned by these functions, called the Besicovitch almost periodic functions.

We can see that $B^2\neq M^2$ since there are functions like

$$ f(x) = \left\{\begin{align}1 \ \ \ \ \ x\ge0 \\ -1 \ \ \ \ \ x < 0\end{align}\right. $$

which is orthogonal to all $e^{i\xi x}$, and $\|f\|_2 = 1$.

Question: I can't seem to find any discussion of $M^p$ independent from $B^p$. Is there a standard name for $M^p$? Is there a convenient description of an orthonormal basis for $M^2$?