How to Show Equicontinuous?

To show that they are equicontinuous, fix any ϵ > 0. Choose N sufficiently large so that N > 2/ϵ. Then for any n>N we have |fn(x) − fn(y)| < ϵ for any x, y. For 1 ≤ n ≤ N, since fn is uniformly continuous on [0,1], there exists δn so that |x − y| < δn implies |fn(x) − fn(y)| < ϵ.

How to prove equicontinuous?

|f(t)|dt < M|x − y|. In any case, if we take δ = ε/M, then |x − y| < δ =⇒ |T[f](x) − T[f](y)| < ε. This shows that T(K) is equicontinuous. To see that the closure is also equicon- tinuous, we use the ε/3 trick.

Is equicontinuous?

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.