Let L be the set of all linear functions on $[0,1]$, $f \in L \Leftrightarrow \exists a,b \in \mathbb{R} \forall x \in [0,1] f(x)=ax+b$
Let $p(f,g) = sup_{x \in [0,1]}|f(x)-g(x)|$. Recall that $C[0,1]$ is the space of all real-valued continuous functions on interval $[0,1]$ with metric p.
1- I need to proof that L is not open in $C([0,1],p)$
2- I have to prove that if $f_n ∈ L$ for all $n \in \mathbb{N}$ and there exists a continuous function f such that for all $x \in [0, 1]$ we have $f_n(x) \rightarrow f(x)$ then f is linear and $ρ(f_n, f) \rightarrow 0$
3- Finally, prove that L is closed in $(C[0, 1], ρ)$, i.e., that if $f_n \in L$ for all $n \in \mathbb{N}$ and there exists a continuous function f such that $ρ(f_n, f) \rightarrow 0$, then f is linear
In the first one I think i need to proof that for every $\epsilon > 0$ and every linear function $f(x) = ax+b$ there exists a continuous function g which is not linear and we have $p(f,g) < \epsilon$, but I'm not sure how to achieve that.
In the second one, coefficients $a_{n}$ and $b_n$ of every linear function $f_n$ are determined by two values of $f_n$, somewhere in [0, 1].
I guess the last one is a combination of the three others
1- If $f$ is affine, let $f_n(x) = ax+b+ {1 \over n} x(1-x)$. Check that $\|f-f_n\| = {1 \over 4n}$.
2- If the $f_n$ are affine and $f_n(x) \to f(x)$, it is straightforward to check that $f$ must be affine. If $a$ is affine it is straightforward to show that $\|a|| = \max(|a(0)|, |a(1)|)$.
3- Define $A$ as follows: $(Af)(x) = f(x) -((f(1)-f(0))x+f(0))$. Check that $Af = 0$ iff $f$ is affine. Note that $\ker A$ is a closed set.