Let G be the group of all real-valued functions on the unit interval [0,1], where we define for f,g elements of G, addition by (f + g)(x) = f(x) + g(x) for every x elements of [0,1]. If N = { f element of G,f(1/4) = 0 }, prove that G/N is ismorphic to real numbers under addition
Hmm....to study this stuff you must have taken before at least an introductory calculuis course and even a little more, and linear algebra and stuff...
Is is true that H(f + g) = H(f) + H(g)? Or what is the same, is it true that (f + g)(1/4) = f(1/4) + g(1/4)? To answer this just check CAREFULLY the definition of addition in G...
About onto: is it true that for any real number r we can find a real valued function defined on [0,1] s.t. f(1/4) = r?
Please do think a little about these questions and make an effort towards their solution, and THEN if you're still stuck write back.
Tonio