Given fx fx1 prove that it is a subspace of CR using the tw

Given f(x) = f(x+1), prove that it is a subspace of C(R) using the two closure properties of scalar multiplication and addition.

Solution

Let S = {f\\f(x) =f( x+1)}

Let us consider two elements belonging to f1 and f2 to S

f1(x) = f1(x+1)

and f2(x) = f2(x+1)

Adding we get

(f1+f2)x=(f1+f2)(x+1)

Hence closure is true.

Next is to check scalar multiplication

f(cx) = f(cx+c)

i.e. scalar multiplication is also closed

So S is a subspace of C(R)