To give some context: I need prove that a family $\mathcal{C}:=\{y: \|y(t)-y(s)\| \leq a(t) - a(s),\ \|y(t)\| \leq b(t)\}\subseteq C([0,T];\mathcal{H})$ is compact, for appropiated $a\in\operatorname{AC}([0,T];\mathbb{R})$ increasing, and $b \in L^{1}([0,T];\mathbb{R})$ functions, that appeared in my calculus of bounds.
I want to use the Arzela-Ascoli theorem, the equicontinuous is direct for the first condition but, I have not been able to prove the relative compactness in the point sense.
It occurs to me that I can try uniform bounding for $\mathcal{C}(t) := \{y(t): y\in\mathcal{C}\}$ but, It's true the next version of Arzela-Ascoli theorem on $C([0,T];\mathcal{H})$?
The next are equivalents:
- $\mathcal{C}$ are relatively compact.
- $\mathcal{C}(t)$ are uniformly bounded and $\mathcal{C}$ are equicontinuous.
My doubt is that I know that it is true when we take $C([0,T];\mathbb{R}^{n})$ (or actually, in the arrival there is a finite dimension).