Let X a metric compact space and Y a Banach Space.
Let f and g $\displaystyle $\in$$ C(X,F)
Prove that: 1) A+B is equicontinuous, and
2) A U B is equicontinuous
I really dont know where to start....
Let X a metric compact space and Y a Banach Space.
Let f and g $\displaystyle $\in$$ C(X,F)
Prove that: 1) A+B is equicontinuous, and
2) A U B is equicontinuous
I really dont know where to start....
Hint:
1. $\displaystyle |f(x_0)-f(x)+(g(x_0)-g(x))|\leq |f(x_0)-f(x)|+|g(x_0)-g(x)| < 2\varepsilon$ if $\displaystyle |x-x_0|<\delta =\min \{ \delta _A, \delta _B \}$ whenever $\displaystyle f\in A, \ g\in B$
2. For the union use the same $\displaystyle \delta$.