# Equicontinuity

• May 15th 2011, 05:26 PM
orbit
Equicontinuity
Let X a metric compact space and Y a Banach Space.

Let f and g $\displaystyle$\in$$C(X,F) Prove that: 1) A+B is equicontinuous, and 2) A U B is equicontinuous I really dont know where to start.... • May 15th 2011, 07:35 PM Drexel28 Quote: Originally Posted by orbit Let X a metric compact space and Y a Banach Space. Let A anb B \displaystyle \in$$ C(X,F)

Prove that: 1) A+B is equicontinuous, and
2) A U B is equicontinuous

I really dont know where to start....

What does this mean? Equicontinuity applies to a family of maps.
• May 15th 2011, 08:52 PM
orbit
Quote:

Originally Posted by Drexel28
What does this mean? Equicontinuity applies to a family of maps.

Hi, A and B are Equicontinuous subsets of C(X,F)
• May 15th 2011, 09:15 PM
Jose27
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 $\displaystyle \delta$.
• May 16th 2011, 06:01 AM
orbit
Quote:

Originally Posted by Jose27
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 $\displaystyle \delta$.

Thank you.

For 2) is the same delta for 1??

Regars.
• May 16th 2011, 07:55 PM
orbit
I finished the exercise.

Thank u!