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....

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- May 15th 2011, 05:26 PMorbitEquicontinuity
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 PMDrexel28
- May 15th 2011, 08:52 PMorbit
- May 15th 2011, 09:15 PMJose27
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 AMorbit
- May 16th 2011, 07:55 PMorbit
I finished the exercise.

Thank u!