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**Linear groups, with an exposition of the Galois field theory**

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12) For any sequence {fn } ⊂ D(A) such that fn and Afn converge boundedly and pointwise to some functions f, g ∈ Cb (RN ), respectively, it holds that f ∈ D(A) and Af = g. Finally, D(A) is dense in Cb (RN ) in the mixed topology. Proof. Fix f ∈ D(A) and x ∈ RN . 3 the right derivative (T (t + h)f )(x) − (T (t)f )(x) d+ (T (t)f )(x) := lim+ dt h h→0 exists at any t ≥ 0 and d+ (T (t)f )(x) = (T (t)Af )(x). 5 the function t → (T (t)Af )(x) is continuous in [0, +∞). 12) holds. Next, let {fn } ⊂ D(A) be as in the statement.

3) in the ball B(n) = {x ∈ RN : |x| < n}. This problem has a unique solution un ∈ C(B(n)) (in Section C we recall the results about elliptic and parabolic problems in bounded domains that we need throughout this chapter). 1), we prove that we can define a function u : RN → R by setting u(x) := lim un (x), n→+∞ for any x ∈ RN . 2) and it satisfies the estimate ||u||∞ ≤ 1 ||f ||∞ . 2) in Dmax (A). It is the unique solution provided further conditions on the coefficients are satisfied. 2) will be treated in Chapter 4.

Finally, D(A) is dense in Cb (RN ) in the mixed topology. Proof. Fix f ∈ D(A) and x ∈ RN . 3 the right derivative (T (t + h)f )(x) − (T (t)f )(x) d+ (T (t)f )(x) := lim+ dt h h→0 exists at any t ≥ 0 and d+ (T (t)f )(x) = (T (t)Af )(x). 5 the function t → (T (t)Af )(x) is continuous in [0, +∞). 12) holds. Next, let {fn } ⊂ D(A) be as in the statement. By the previous step, for any x ∈ RN and any n ∈ N, the function (T (·)fn )(x) is differentiable in [0, +∞) and d (T (s)fn )(x) = (T (s)Afn )(x), s ≥ 0.

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