By Hatcher A.

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**Example text**

Of functions fm : S → Y converges uniformly to a function f : S → Y if, for every positive real number ε there is an integer m such that dY (f (x), fp (x)) < ε, for p > m and all x ∈ S. A sequence f0 , f1 , . . satisfies the Cauchy criterion if, for every positive real number ε, there is an integer m such that dY (fp (x), fq (x)) < ε, for p, q > m, and all x in S. (a) Show that a sequence f0 , f1 , . . of functions fm : S → Y that converges to a function f : S → Y , satisfy the Cauchy criterion.

With sm (x) = |i|≤m ci xi converges uniformly in P (0, r ) for all r < r. In particular i∈I ci xi converges in P (0, r). Indeed, we have that i |ci |r = i∈I |ci |r i∈I ir i ri ≤C i∈I ri =C ri n (1 − i=1 r i −1 ) . 6. Let U be an open subset of Kn . A function g: U → K is analytic in U if, for each x in U , there is an r in R and a formal power series f (x) = i i∈I ci x which is convergent in P (0, r), such that g(x + h) = f (h) for all h ∈ P (0, r) such that x + h ∈ U. A function g = (g1 , . .

The left translation λA induces a homeomorphism λA |U : U → λA (U ) of metric spaces onto the open neighborhood λA (U ) of A, with inverse λA−1 . Consequently, for each A, we have a homeomorphism ϕA : V → UA onto some neighborhood of A.