I am trying find that below set is equicontinuous or not:
$$K=\{f\mid f\in \Pi_n , \|f\|\le 1\}$$ $$\Pi_n=\{\text{polynomials of degree }\le n \text{ over } [a,b]\}$$ with norm : $$||f||=\sup_{x\in \mathcal{D}(f)}|f(x)|$$ I proved that if we change the condition $f\in \Pi_n$ to $f\in \mathcal{C}$ it isn't equicontinuous with use of Weierstrass theorem:
$$W_\varepsilon:\mathcal{C} \to \Pi \quad st \quad \|W_\epsilon(f)-f\|<\epsilon$$
So for every $0 < \varepsilon <1$ and $0<\delta$ we have the function $f_\delta(x)=W_\frac{1}{5}\left(\frac{4}{5}\sin(n\pi x)\right)$ where $\frac{1}{n}<\delta$ st equcontinuity doesn't hold:
$$\left|0-\frac{1}{n}\right|<\delta\text{ but }\left|f_\delta(0)-f_\delta\left(\frac{1}{n}\right)\right|>\varepsilon$$
Which side of conjecture is true?