Given: vector V={f:[0,1]-R} with addition: (f+g)(x)=f(x)+g(x) where x is an element of [0,1], and scalar multiplication: (cf)(x)=c(f(x)) where x is an element of [0,1], c is scalar in R, have to prove that certain sets are subspaces of the vector. For example f(1)=2f(0)-f(1/2)... how do i prove that its not empty/contains 0? Can you explain briefly also what's going on, how the functions fit in? Thanks!

Question

Given: vector V={f:[0,1]->R} with addition: (f+g)(x)=f(x)+g(x)  where x is an element of [0,1], and scalar multiplication: (cf)(x)=c(f(x)) where x is an element of [0,1], c is scalar in R, have to prove that certain sets are subspaces of the vector. For example f(1)=2f(0)-f(1/2)... how do i prove that its not empty/contains 0? Can you explain briefly also what's going on, how the functions fit in? Thanks!