Question

If f is a continuos map from a normed space x to y and e is a totally bounded subset of x .Prove that f(e) is totally bounded ij y

Answer 1

First X is not necessarily compact. For example X=(0,1)⊂R is a totally bounded space but is not compact.

To prove that f(X) is totally bounded, you have to prove that for any ϵ>0, f(X) is contained in the union of a finite number of balls of radius ϵ.

So take ϵ>0. As f is uniformly continuous, you can find δ>0 such that dX(x,y)<δ implies dY(f(x),f(y))<ϵ. As X is supposed to be totally bounded, you can find a finite number of balls BX(a1,δ),…,BX(an,δ) such that

X⊂∪ni=1BX(ai,δ)

but then

f(X)⊂∪ni=1BY(f(ai),ϵ).

I hope this will help you

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