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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?

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  • $\begingroup$ Thank you! I had the same question reading the same book by Kreyszig. Though I knew that the Schauder basis is countable, the difference/ no "continuity" in terminology confused me. Maybe we could say that a total family if it is countable is called a "Schauder basis", and "Hammel basis" if it is finite. We would then have total families that can be uncountable, countable or finite. Or call a total family a "potentially uncountable schauder basis". Anyway maybe I should not propose new terminology, but I think a cross reference at Kreyszig's book between these two definitions would be great. $\endgroup$
    – Adonis Bozis
    Commented May 28, 2020 at 11:05

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So does the author just mean that the basis is not a Hamel basis?

Yes, precisely that; "a basis, in the sense of algebra" is a Hamel basis.

I think that a total orthonormal sequence must be a Schauder basis

Yes, that is correct, but a

total orthonormal family

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.

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    $\begingroup$ Note that a Hilbert space admits a countable orthonormal basis if and only if it is separable. $\endgroup$
    – Math1000
    Commented Jan 23, 2015 at 20:33
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    $\begingroup$ Thanks! that last remark about the countability is exactly the kind of thing that I felt I wasn't seeing. Is there some 'classic example' of an uncountable orthonormal basis for a well known space like $\mathbb{L}_2$? $\endgroup$
    – user2520938
    Commented Jan 23, 2015 at 20:34
  • $\begingroup$ @Math1000 This is a 'consequence of' the gram schmidt process right? $\endgroup$
    – user2520938
    Commented Jan 23, 2015 at 20:39
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    $\begingroup$ @user2520938 The usual $\mathbb{L}_2$ spaces for the Lebesgue and closely related measures are separable. For other measures, you can have inseparable spaces. The most straightforward example of an inseparable Hilbert space is $$\ell^2(S) := \left\{ f \colon S \to \mathbb{C} \;\Big\vert \sum_{s\in S} \lvert f(s)\rvert^2 < +\infty \right\}$$ for an uncountable set $S$, which is the space of square-integrable functions on $S$ with respect to the counting measure. The family $\{ \delta_s : s \in S\}$, where $$\delta_s(t) = \begin{cases}1&,s=t\\0&,s\neq t \end{cases}$$ is a Hilbert basis. $\endgroup$
    – Daniel Fischer
    Commented Jan 23, 2015 at 20:39

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