The space $C[a,b]$ , space of all real valued continuous functions on $[a,b]$ is an infinite dimensional vector space over the field $\Bbb R$. As every vector space over a field has a basis so definitely $C[a,b]$ has a basis. I want to know a basis of $C[a,b]$. How I can find a basis of it ?
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$\begingroup$ Fourier series? $\endgroup$– BAICommented Nov 4, 2017 at 14:48
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$\begingroup$ @BAI What do you mean ? $\endgroup$– EmptyCommented Nov 4, 2017 at 14:50
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1$\begingroup$ A (Hamel) basis cannot be described explicitly. For a Schauder basis you need to specify a norm. $\endgroup$– egregCommented Nov 4, 2017 at 14:52
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1$\begingroup$ @egreg Ok. So if $\sup$ norm is given then what's the Schauder basis ? $\endgroup$– EmptyCommented Nov 4, 2017 at 14:54
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$\begingroup$ I'm not getting why negative vote ? $\endgroup$– EmptyCommented Nov 4, 2017 at 15:05
2 Answers
You mention that every vector space has a basis, which is a fact about Hamel bases, but your question is tagged with shauder-basis. Which one are you asking about? I'll just answer both and highlight the differences.
A Hamel basis is what we usually mean when we just say "basis". I.e. every element can be uniquely written as a finite linear combination of basis elements. Necessarily, any Hamel basis of $C[a,b]$ will be uncountable. See What is a basis for the vector space of continuous functions? for why there is essentially no way to write down a Hamel basis of $C[a,b]$ without invoking the Axiom of Choice.
A Schauder basis is a sequence $(f_n)$ such that for any element $f$ of the space, there is a unique sequence of constants $(\alpha_n)$ such that $f = \lim_n \sum_{i=1}^n \alpha_n f_n$, where the limit is taken with respect to the norm of the space (in this case, the sup norm on $C[a,b]$). The most common Schauder basis for $C[0,1]$ is a certain type of wavelet basis, called a Faber–Schauder system. Describing such a system is a task that I don't attempt here, but I encourage reading the linked wiki.
If you're looking for a Schauder basis, meaning a basis from which every vector may be written as an infinite sum, then by the Stone-Weierstrass theorem, the polynomials are dense in $C[a,b]$ in the supremum norm. That means every continuous function can be written as an infinite sum of polynomials. For example Bernstein polynomials are a convenient spanning set. But if you don't like polynomials, many other families of functions also serve. The Stone-Weierstrass theorem applies to any family of functions that separates points. Trig functions, exponentials, absolute values, whatever you want. And there is a process for turning these spanning sets into bases.
But if you want an actual algebraic basis in the sense of linear algebra, a Hamel basis, a basis such that every vector is a finite linear combination, well such things exist by the axiom of choice, but this is nonconstructive. No explicit formula for such bases may be given in general, and it is even consistent with ZF set theory that they not exist.
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2$\begingroup$ The polynomials do span $C[a,b]$ in the Shauder sense, but they are not a basis. $\endgroup$– nullUserCommented Nov 4, 2017 at 14:59
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$\begingroup$ @nullUser, I believe this distinction is already addressed in my answer, or do you think it is unclear? $\endgroup$ Commented Nov 4, 2017 at 15:02
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$\begingroup$ your answer seems to imply that the Bernstein polynomials, trig functions, absolute values are all Shauder bases of $C[a,b]$, when in reality none of them are. For instance, the Bernstein polynomial are called "Bernstein basis polynomials" because the first $n$ of them constitute a (Hamel) basis of the polynomials of degree at most $n$. It is not clear to me that this is what you meant from your post, and if it is what you meant, then you still haven't answered OPs question because OP asked about a basis, not just spanning sets. $\endgroup$– nullUserCommented Nov 4, 2017 at 15:07
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$\begingroup$ It is interesting that it is consistent with ZF that no explicit formula for a Hamel basis exists, I did not know this. $\endgroup$– nullUserCommented Nov 4, 2017 at 15:10
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$\begingroup$ @nullUser Asaf's answer in the question you linked in your answer shows why there is no Hamel basis in Solovay's model. $\endgroup$ Commented Nov 4, 2017 at 15:18