Errata for Fundamentals of Advanced mathematics — Volume 2

Errata for Fundamentals of Advanced mathematics — Volume 2 1. p. 12, line 6, read "K′ (α′ )" instead of "K (α′ )". 2. p. 17, line 19, read "=" instead of "="; line 21, read "does not exist" instead of "exists". 3. p. 20, line 11, read "0 ≤ i ≤ r" instead of "1 ≤ i ≤ r". ¯ instead of "x ∈ K". 4. p. 24, line 28, read "x ∈ K" ¯ instead of "K". 5. p. 27, line 10, read "K" 6. p. 32, line 3, read "a is not of the form bn , b ∈ K" instead of "an ∈ / K"; line 5, read "ζ" instead of "ς". 7. p. 43, line 10, after "yields", add "with j = 0, ..., n − 2"; in the last sum (j) (j+1) of line 11, read "ui " instead of "ui ". 8. p. 51, replace the sentence beginning line 23 by: "The coefficients c and d are free parameters, the former in C, the second in C× ; indeed, putting 2 M = C t, e−t /2 we have that GalD (N, M) = C, GalD (M, K) = C× , which yields the short exact sequence of Abelian groups 0 −→ C −→ G −→ C× −→ 0, in other words G is an extension of C× by C ([P1], section 2.2.2(II)).". 9. p. 62, lines 27, 29; p. 63, lines 3, 8; p. 64, lines 7, 9: read " x′j instead of "

j∈J

"

x′j i∈J ".

10. p. 83, line 22, read "smallest" instead of "largest" and "=" instead of "=". 11. p. 91, line 28, read "j

i0 " instead of "j ≥ i0 ".

12. p. 92, line 19, add after the last sentence: "This extension is then unique.". 13. p. 95, line 8, read "∀ (x′ , x′′ )" instead of "∀ (x′ ; x′′ )". 14. p. 105, line 25, read "∀i

i′′0 " instead of "∀i

i0 ".

15. p. 119, line 17, read "K" instead of "K". 16. p. 128, lines 1, 2, read "(iv)" instead of the second "(iii)" and "(v)" instead of "(iv)". 17. p. 132, line 20, read "ξx0 → ξ" instead of "ξ → ξx0 ". 18. p. 133, line 14, delete "in Lcsh".

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19. p. 135 lines 19-21, read "reduced if for all i ∈ I, the projection pri (E) , where pri is the canonical projection i Ei ։ Ei , is dense in Ei " instead of "decreasing [...] Tj )"; lines 27-31, replace Statement 2) of Remark 3.33 by: "Every projective limit can be put into the form of a reduced projective ˜ j is the restriction of ψj to limit: if Fi = pri (E) and ψ i i i Fi , then E is j ˜ ker ψ of Fi .". the subspace i j

i

20. p. 137, last line, read " p

i

n i=1

xi p " instead of " p

n i=1

xi

P

".

21. p. 141, lines 24, 26, delete "decreasing". 22. p. 142, lines 10, 11, delete "is surjective (ibid.) and". 23. p. 144, line 11, read "bounded in F " instead of "bounded in M ". ¯ 24. p. 168, line 13, read "¯ x ∈ A" instead of "¯ x ∈ R". 25. p. 150, line 6, delete "decreasing"; line 8, read "mapping" instead of "surjection" and "→" instead of "։". 26. p. 172, line 25, read "exact" instead of "quasi-exact"; delete "and from FSop to Sil" and Note 17. 27. p. 173, line 2, delete "strict"; line 5, read "an" instead of "a strict"; line 7, delete "strict" and read "Silva" instead of "(FS)-"; line 8, read "−→" instead of "֒→" and "mapping" instead of "injection"; line 8, read "mapping" instead of "surjection", "−→" instead of "֒→" and delete "strict"; line 10, read "(FS)-" instead of "Silva"; replace lines 13-27 by: "(1) Since the spaces (Ei′ )b are Fréchet, they are bornological, and their inductive limits as well. Thus they are Mackey spaces and the result follows from ([SCF 99], Chapter IV, Section 4.4). (2) The Ei are reflexive (DF) spaces, and the result follows from ([SCF 99], Exercise 24(f), p. 197).". 28. p. 174, line 2, delete "increasing"; line 3, read "mapping" instead of "injection" and "−→" instead of "֒→". 29. p. 175, line 14, read "L (R; S)" instead of "L (R, S)". 30. p. 177, lines 8, 9, read "

u∈H,x1 ∈A1

u (x1 , A2 )" instead of "

u∈H,x1 ∈A1 ".

31. p. 185, line 5, read "such" instead of "any"; lines 15, 23, read "≥" instead of "≤". 32. p. 186, line 11, read "| ei |x |2 " instead of " ei |x

2

".

33. p. 188 line 27, add "salient" after "closed". 34. p. 193, line 12, add ", v = 35. p. 204, line 21, read " p

r ′ i=1 xi

⊗ yi " at the end of the line.

|f |p .dµ" instead of " p 2

|f | .dµ".

36. p. 218, 3rd line of the proof, read " of " h1 |f (x + h) − f (x)|".

1 h

(f (x + h) − f (x)) − g (c) " instead

37. p. 230, line 16, add "nonzero" after "many". 38. p. 253, lines 1, 3, delete "strict"; line 26, add "denoted by Cr (0)" after "plane". 39. p. 299, line 18, read "is an A-module" instead of "is the A-module W ". 40. p. 204, line 13 read "equation" instead of "solution".

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