Prove the set of functions such that 02fxdxf1 is a subspace

Prove the set of functions such that 02f(x)dx=f(1) is a subspace or not?

Prove the set of functions such that 02f(x)dx=1 is a subspace or not?

Solution

A subset U of a vector space V is called a subspace of V if the following two conditions hold:
S1) If u, u0 are any two vectors in U then u + u0 U.
S2) If u U and c R, then cu U.
In brief, U is a subspace if it is closed under addition and scalar multiplication.

Here the interval is [0 2]

since ₀²f(x)dx = 1

and 1 [0 2]

So, the set of functions such that ₀²f(x)dx=1 is a subspace.