Subspace of Vector Space Given

Question

Suppose $W = \{f \in C[1,2] | \int_{1}^2 f(x)~dx = a\}$. If $W$ be the subspace of $C[1,2]$ then what is the value of $a$?

Do you have any idea for this? I don't found information about $C[1,2]$ expression in the textbook. I want to apply the theorem of subspace, but seems will lead to trouble. Please help.

Answer 1

Guide:

A subspace is closed under addition and scalar multiplication.

Suppose $\int_1^2 f(x) dx = a$, we need $\int_1^2 2f(x) dx=a$, subtract them and you should be able to see the value of $a$.

Answer 2

The zero vector in $\mathscr C[1, 2]$ is the function $z:[1, 2]\to\Bbb R$ given by $z(x)=0$.

If $W$ is a subspace, then $z\in W$ so that $$ a=\int_1^2 z(x)\,dx=\int_1^20\,dx=0 $$