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anonymous
 one year ago
Prove that the space of realvalued 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.
anonymous
 one year ago
Prove that the space of realvalued 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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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0your awesome if you know this!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well, 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0clearly 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the hard part is finding a basis, though; this is probably cheating but it is wellknown that Haar wavelets work here

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0https://en.wikipedia.org/wiki/Haar_wavelet#Haar_system_on_the_unit_interval_and_related_systems
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