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Consider $X = mathcal{C}([−1,1])$ with the usual norm $|f|_{infty} = sup_{tin [−1,1]}|f(t)|.$

Define
$$mathcal{A}_{+}={ f in X : f(t)=f(−t) space forall tin [−1,1]},$$
$$mathcal{A}_{−}={ f in X : f(t)=−f(−t) space forall t in [−1,1]}. $$

Is $mathcal{A}_{+} +mathcal{A}_{−} = {f +g : f in mathcal{A}_{+},g in mathcal{A}_{−}}$ meager?

I know this set is dense by the Stone-Weierstrass Theorem. However, that doesn't really help. I also know that if the set is closed, then it is meager, but I have difficulties deciding whether it is closed or not. I know the exponential function is a limit of a sequence of a sum of even and odd functions, however one could define it to be that, in which case it doesn't help.

Any hints on how to get going on this problem, and on whether the set $mathcal{A}_{+}+{A}_{-} $ is closed or not? Thank you in advance.

Note that any function can be written as
$f(x) = {1 over 2} (f(x) + f(-x)) + {1 over 2} (f(x) - f(-x)) $, so
$mathcal{A}_{+} +mathcal{A}_{−} = X$, which is not meagre.

(It is not meagre because $C[-1,1]$ is a complete metric space.)