# Equicontinuity

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

Let f and g $\in$ 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 $\in$ 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. $|f(x_0)-f(x)+(g(x_0)-g(x))|\leq |f(x_0)-f(x)|+|g(x_0)-g(x)| < 2\varepsilon$ if $|x-x_0|<\delta =\min \{ \delta _A, \delta _B \}$ whenever $f\in A, \ g\in B$

2. For the union use the same $\delta$.
• May 16th 2011, 06:01 AM
orbit
Quote:

Originally Posted by Jose27
Hint:

1. $|f(x_0)-f(x)+(g(x_0)-g(x))|\leq |f(x_0)-f(x)|+|g(x_0)-g(x)| < 2\varepsilon$ if $|x-x_0|<\delta =\min \{ \delta _A, \delta _B \}$ whenever $f\in A, \ g\in B$

2. For the union use the same $\delta$.

Thank you.

For 2) is the same delta for 1??

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

Thank u!