Prove that the space of real-valued continuous functions defined on the interval [0, 1], C^0 [0, 1], is a vector space over the real scalars, and find a basis for this space.

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Prove that the space of real-valued continuous functions defined on the interval [0, 1], C^0 [0, 1], is a vector space over the real scalars, and find a basis for this space.

Mathematics
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your awesome if you know this!
well, this boils down to showing that \(f,g\) are continuous gives \(af+bg\) is continuous, i.e. closure of \(C^0([0,1])\) under linear combinations
clearly we have additive inverses \(f\mapsto -f\) since \(f+-f=0\) and the properties like associativity and scalar multiplication are all inherited from those of addition and multiplication of normal real expressions

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the hard part is finding a basis, though; this is probably cheating but it is well-known that Haar wavelets work here
https://en.wikipedia.org/wiki/Haar_wavelet#Haar_system_on_the_unit_interval_and_related_systems

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