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.

Question

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.

anonymous · Answer

your awesome if you know this!

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

the hard part is finding a basis, though; this is probably cheating but it is well-known that Haar wavelets work here

anonymous · Answer

https://en.wikipedia.org/wiki/Haar_wavelet#Haar_system_on_the_unit_interval_and_related_systems