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I can define a Hilbert space that is :

1) [tex]\mathbb{R}^n[/tex] with the euclidian norm, especially on a real field, and which is finite dimensional : is it right ? This is the most stupid question ever.

2) over the quaternions [tex]\mathbb{H}[/tex] ?

3) if the dimension is infinite non-countable, it is not separable. There is no need to talk about topological or metrical separability, the two coincide.

Please some one answer. It is due to questions [thread=44301]here[/thread].