Question

show that the set of all continues function in the interval of [-1,1], c^0([-1,1]) is a vector space

Answer 1

$f \in C^0$

$f:[0,1]\to[0,1]$

$\forall f\in C^0 \text{ by the condition of the problem }$

$f \text{ is continues function}$

$\text{By the properties of continuous functions:}$

$\text{Sum and scalar multiples of continuous functions are also }$

$\text{continuous (and addition is commutative)}.$

$0 \in C^0; f+0 =f$

$\text{That is, }C^0 \text{ fully supports the axiomatics of the vector space}$

$\text{Hence }C^0 \text{is a vector space of continuous functions}$