I'm learning about Hilbert spaces and related things from the book "Introductory functional analysis with applications". Now I just read the following sentence, which I don't quite understand:
"A total orthonormal family in $X$ is sometimes called an orthonormal basis for $X$. However, it is important to note that this is not a basis, in the sense of algebra, for $X$ as a vector space, unless $X$ is finite dimensional."
But I think that a total orthonormal sequence must be a Schauder basis, basically just from the definition. So does the author just mean that the basis is not a Hamel basis? Or is there something more subtle going on here that I'm not seeing?
Best Answer
Yes, precisely that; "a basis, in the sense of algebra" is a Hamel basis.
Yes, that is correct, but a
need not be countable, in contrast to a sequence, and if you have an uncountable total orthonormal family in a Hilbert space $H$, that is a Hilbert basis of $H$, but not a Schauder basis, since Schauder bases are by definition countable.