Let S f C02f1 1 set of continuous real valued functions de

Let S = {f C[0,2]|f(1) = 1} set of continuous real valued functions defined on [0,2]. Show that S is not a vector space by giving a vector space axiom that fails to hold.

Solution

For S to be a vector space we need that for any ,f and g in S,f+g must also be in S

(f+g)(1)=f(1)+g(1)=1+1=2

Hence, f+g is not in S if f and g are in S

Hence, S is not a vector space.