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What are examples of a basis of $C[0,1]$? (Hamel basis or Schauder basis... related: What is the difference between a Hamel basis and a Schauder basis?)

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Do you mean as a vector space over $\mathbb{R}$ ? –  WLOG Feb 7 at 11:27
    
The usual one is described here. Schauder discovered this basis, I believe; it is called the Schauder Basis of $C[0,1]$. –  David Mitra Feb 7 at 11:28
    
    
Do you mean a Hamel basis? –  Matemáticos Chibchas Feb 7 at 12:24

2 Answers 2

I assume you want Hamel basis. If you mean Schauder basis (or some other type of basis, like a base for the topology), you should say so.

I cannot write down a Hamel basis for $C[0,1]$. However, I can write down a linearly independent set of cardinal $\mathfrak{c} = 2^{\aleph_0}$. For $s \in (0,1)$, let $g_s \in C[0,1]$ be piecewise linear with $g_s(0)=0$, $g_s(1)=0$, $g_s(s) = 1$. The uncountable set $\{g_s : s \in (0,1) \}$ is linearly independent.

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The Faber-Schauder basis of $C[0,1]$, mentioned by David Mitra, consists of the following functions: $f_0\equiv 1$, and after that $$f_{j,k} = (1-2^j|x-k/2^j|)^+,\quad j\ge 0, \quad 1\le k\le 2^j, \ \text{ $k$ is odd}$$ Apart from $f_0$, these are essentially antiderivatives of Rademacher functions, which hints at the reason for their good basis properties. The form a monotone Schauder basis for $C[0,1]$. In contrast, polynomials and trigonometric polynomials fail at this task.

A couple of pictures: functions up to $j=2$

basis

and up to $j=3$

basis2

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