By Boij M., Laksov D.

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Indeed, this is clear for n = 1 and follows in the general case by induction on n. Let r < r and let C = i∈I |ci |r i . Then i |ci r | ≤ C Conversely, given a formal power series for all i ∈ I. i∈I ci xi , such that |ci |ri ≤ C, for some C, then s1 (x), s2 (x), . . 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 .

Then we have that Φ(exp(iy)) = exp(Φ(iy)). In the last formula the exponential function on the left is the usual exponential function for complex numbers and the one to the right the exponential function for matrices. 6. The exponential function defines a continuous map exp : Mn (K) → Mn (K). Indeed, we have seen that expm (X) ≤ exp( X ). Let B(Z, r) be a ball in Mn (K), and choose Y in Mn (K) such that Z + r ≤ Y . Then, for any X in B(Z, r), we have that X ≤ X − Z + Z ≤ r + Z ≤ Y . Consequently, we have that expm (X) ≤ exp( X ) ≤ exp( Y ), for all X in B(Z, r).

4 we shall show that the 2 logarithmic function is analytic. Hence, in particular, it is differentiable with an analytic derivative. 1. Determine the matrices exp ( 10 11 ), exp ( 14 11 ), and exp 1 1 −1 3 . 2. Show that in a metric space every convergent sequence is a Cauchy sequence. 3. Let X be a set, S a subset, and (Y, dY ) a metric space. A sequence f0 , f1 , . . 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.

### An introduction to algebra and geometry via matrix groups by Boij M., Laksov D.

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