$W=\left\{f\mid f(3)-f(-5)=0\right\}$ is vector space or not?

Question

Is $W=\left\{f\mid f(3)-f(-5)=0\right\}$ a vector space or not?

Above example is the vector space over a field of real ?

Answer 1

HINT

Think to the key properties for a vector space, notably

1. $\vec{0} \in W \implies$ does $f_0(x)=0 \,\forall x$ fulfil this condition
2. $\vec{cv}\to c \cdot \vec{v}\in W\implies$ suppose $f(3)-f(-5)=0$ then what about $c\cdot f$?
3. $\vec{v}+\vec{w} \in W\implies$ let $f_1(3)-f_1(-5)=0,\,f_2(3)-f_2(-5)=0$ then what about $f_1+f_2$?

Answer 2

Hint:

If we define the operations in the vector space as the usual pointwise addition and scalar multiplication, for two functions $f,g \in W$ and two scalars $a,b$ we have:

$$ (af+bg)(3)-(af+bg)(-5)=af(3)+bg(3)-af(-5)-bg(-5)= $$ $$ =a(f(3)-f(-5))+b(g(3)-g(-5))=a\cdot 0 +b\cdot 0=0 $$