Question

Suppose a is element of R. Show that the set of continuous real-valued functions f on the interval [0;1] such that integral f from 0 to 1 = a is a subspace of R[0;1] if and only if a = 0

Answer 1

Suppose $a\in R$ and $\int_0^1f=a$ is a subspace of $R^{[0,1]}$ then we have this integral will be subspace whenever for some 'b'

Such that $b.a=a.b=0,$ hence a=0

Conversely if $a=0$ , then the are under the curve between the interval [0,1] bounded the curve of functions f is zero. Hence Set of functions form a subspace.