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Algebraic Theory of D-modules [Lecture notes] by J. Bernstein

By J. Bernstein

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Extra resources for Algebraic Theory of D-modules [Lecture notes]

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C, ~(E, E')) , 32 E = C - C if and E' (E, C, 2 ) is an artier-convex subspace of is locally decomposable, Proof. (a) , ~herefore (b) ~F(E, E') ~F(E, E') . Let I] L • = L• L• on [0, I] ~(L • L~) . Let V constanh f'anction I (as an element in L~ . C O. E') . ~D(E, E') equipped with the usual ~(L • L =) , be the polar in and Z L• and is of hhe =he closed unit ball Therefore of . m Li . 4) Let decomposable Proposition. \$~ace. ~-eJLos~e of decomposable space. Denote big C i__n ~ . Since Z is not a must not be a neighborhood is no~ a locally decomposable but it is well ~nown that the ~E, It is well known that the n(rm L ~) denotes the positive cone in ~(L I , L~)-neighborhood af we It is easily seen /qat V~C-VnCC2Z Where For the sufficiency, is precisely the Mackay topology strictly finer than .

The second assertion is also a consequence of Schaefer's theorem. 10) Proposition. Let (E, C, 2) be an infrabazu~lled, ordered convex space with a countable fundamental system of 2-bounded subsets of Then (E, C , 2 ) is locally decomposable is locally o-convex and Proof. E' (E', C', #(E', E)) any (E, C, 2) E* . E~ . is locally decomposable. ~e,1. y if E . B 2-equicontinuous. 1) (b) , be a #(E', E)-bounded convex subset of Since so (E, C, 2) is locally decomposable, F ( ~ ) = (~ + c ' ) n (~ - c , ) F(B) E' .

Into C (E, 9) is in the proof of Proposition J is nearly open. 36]), J is an open mapping; in otl~r words, for any convex circled ~-neighbourhood V ~ 0 in E , 2-neighbourhood W of 0 in E that G = E and there exists a c ~ e x such that 2D = 2 " Therefore W C D(V) , (E, C, 2) circled it then follows is a complete, locally decomposable space, and the proof is complete. Remark. 10) Theorem. is a ~e~eratin~, then Proof. ~-cone in For ~ (E, II'II) ~ theorem. ~) 3(v) ~ F(v) , inequality is true since v~ n c - v ~ C V .