Timeline for Question about Schauder bases in C([0,1]).
Current License: CC BY-SA 4.0
14 events
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| Nov 7, 2023 at 13:44 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Mar 21, 2021 at 16:04 | history | edited | Pietro Majer | CC BY-SA 4.0 |
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| Nov 1, 2010 at 13:40 | comment | added | Pietro Majer | Nice notes. As to example 1, it is maybe worth quoting the Müntz–Szász theorem, that characterizes which subsets of monomials are total in $C[a,b]$ (see e.g. en.wikipedia.org/wiki/M%C3%BCntz%E2%80%93Sz%C3%A1sz_theorem). In particular, there is no minimal total family of monomials. Also, proving that e.g. monomials with powers in a given arithmetic sequence are a total family may be a nice exercise. | |
| Nov 1, 2010 at 12:27 | vote | accept | Andrew Stacey | ||
| Nov 1, 2010 at 10:40 | comment | added | Andrew Stacey | Okay, I'm accepting this answer for the statement: "if f admits a representation as uniform limit of a series $\sum_k c_k e^{i k t}$ then the series is its Fourier series". That's the bit I hadn't grokked. More generally, I was so used to being able to pass back and forth between sequences and series that the difference between a topological basis (sequences) and Schauder basis (series) wasn't evident. Thanks for helping me see the light! I've tried to summarise what I've learnt at: ncatlab.org/nlab/show/basis+in+functional+analysis no doubt I've made yet more elementary errors. | |
| Nov 1, 2010 at 8:48 | comment | added | Andrew Stacey | That's actually my reason for wanting to get this straight: what I'm really trying to tell my students is: "Hilbert spaces are great!" and one thing I want to say is that in a Hilbert space we don't have to worry about this mess - but I wanted to be sure that I didn't lie about the actual mess. As for "total", that doesn't include "linearly independent" so isn't the name I was looking for. | |
| Oct 31, 2010 at 22:53 | comment | added | Pietro Majer | Yes; and talking about the distinction "topological vs Schauder basis", it is worth observing that for orthonormal families, the two notions coincide (that's the magic of the Hilbert structure). As to the name, the term "total" is sometimes used for a set whose linear span is dense. | |
| Oct 31, 2010 at 18:18 | comment | added | Andrew Stacey | (ctd) but I think that that is because of the extrinsic meaning of Fourier series. If one said simply, "The dual basis" then I probably wouldn't have had the confusion. Thanks. | |
| Oct 31, 2010 at 18:16 | comment | added | Andrew Stacey | Okay, I think I'm beginning to see a glimmer of light. The monomials are linearly independent and have dense span (is there a name for that?). The trig polys are a topological basis. But the distinction between topological and Schauder bases is that for the former, a sequence each of whose terms is a finite sum converges to the limit, but for a Schauder basis, there is a series which converges. I'm used to thinking of sequences and series as effectively the same thing so I didn't see that there is a difference in this case. I still find the focus on Fourier series distracting (ctd) | |
| Oct 30, 2010 at 22:29 | comment | added | Hany | The usual example in $C([0,1])$ is $\vert t-\frac 12\vert$. Its Fourier series does not converge uniformly. | |
| Oct 30, 2010 at 20:58 | comment | added | Pietro Majer | Exact; e.g. if $P_n$ is a sequence of polynomials uniformly converging to $|t|$ on $[-1,1]$, the coefficients would vary like crazy! | |
| Oct 30, 2010 at 20:50 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Oct 30, 2010 at 20:30 | comment | added | Andrew Stacey | Hmmm. Maybe the point where I'm confused is earlier than I thought. If I take a sequence of polynomials converging uniformly to some continuous functions, then from what you say here then I have to conclude that there is no way (in general) to make that sequence into a series in which the coefficients of each monomial converge. | |
| Oct 30, 2010 at 20:22 | history | answered | Pietro Majer | CC BY-SA 2.5 |