The paper I am reading currently induced some thought about finding a reflexive but non-separable normed space.
In fact the problem can be reduced to finding a non-separable Hilbert space since any Hilbert space is reflexive, which can be proved by so called “Riesz-Frechet theorem” or Riesz representation theorem.
It is said that they are studied by physicists as some such spaces are useful to them. However, they admit no countable complete orthonormal sets. Well…
Now I am thinking of a space of real valued functions on the real line, such that each function vanishes at all but countably many points, and the values of it are square-summable. Assign to the most natural inner product. I will check whether it is a candidate.
Of course I am also interested in non-separable reflexive space that the norm cannot be induced from any inner product.
Welcome any comments and suggestions. I try to add more details later.