TY - JOUR
T1 - Erratum to "The product on smooth and generalized valuations''
T2 - The product on smooth and generalized valuations (Amer. J. Math., (2012) 134, 2, (507-560))
AU - Alesker, S.
AU - Bernig, A.
N1 - Publisher Copyright: © 2016 by Johns Hopkins University Press.
PY - 2016
Y1 - 2016
N2 - Corrections to: S. Alesker and A. Bernig, The product on smooth and generalized valuations, Amer. J. Math. 134 (2012), no. 2, 507-560. In Theorem 8.3 [1] it was incorrectly stated that the product on generalized valuations with given wave front sets is a separately continuous bilinear map between appropriate linear topological spaces. In fact instead of topological continuity one has a weaker property of sequential continuity. On the other hand, the separate (sequential) continuity can be replaced by a stronger property of joint (sequential) continuity. Thus the correct statement, in the notation and assumptions of Theorem 8.3, is that the product of generalized valuations (Formula Presented) is a jointly sequentially continuous bilinear map. Explicitly that means that if one has two sequences (Formula Predented). Appropriate corrections are required in a few other places as well. However the difference between sequential and topological continuity is not significant for this paper and does not influence the rest of the paper. The complete list of corrections is below, including typos we noticed. • p. 516, Proposition 2.7: “linear continuous map” is replaced with “linear sequentially continuous map”. • p. 517, Proposition 2.9: “bilinear separately continuous map” is replaced with “bilinear jointly sequentially continuous map”. p. 518, Proposition 2.13: “a continuous linear map” is replaced with “a sequentially continuous linear map”. • p. 519, line -1: “a separately continuous bilinear map” is replaced with “a jointly sequentially continuous bilinear map”. • p. 520, lines 1,2: “continuous” is replaced with “sequentially continuous”. • p. 550, Theorem 8.3 (line 1): “separately continuous bilinear map” is replaced by “jointly sequentially continuous bilinear map”. • p. 552, line 7: “separate continuity” is replaced with “joint sequential continuity”. • p. 552, line 9: “continuity” is replaced with “sequential continuity”. • p. 553, lines 8 and 10: “by continuity” is replaced with “by sequential continuity”. • p. 553, line 14: “separate continuity” is replaced with “joint sequential continuity”. • p. 553, line 15: “property (4)” is replaced by “property (3)”. • p. 553, line 15: “is dense” is replaced by “is sequentially dense”.
AB - Corrections to: S. Alesker and A. Bernig, The product on smooth and generalized valuations, Amer. J. Math. 134 (2012), no. 2, 507-560. In Theorem 8.3 [1] it was incorrectly stated that the product on generalized valuations with given wave front sets is a separately continuous bilinear map between appropriate linear topological spaces. In fact instead of topological continuity one has a weaker property of sequential continuity. On the other hand, the separate (sequential) continuity can be replaced by a stronger property of joint (sequential) continuity. Thus the correct statement, in the notation and assumptions of Theorem 8.3, is that the product of generalized valuations (Formula Presented) is a jointly sequentially continuous bilinear map. Explicitly that means that if one has two sequences (Formula Predented). Appropriate corrections are required in a few other places as well. However the difference between sequential and topological continuity is not significant for this paper and does not influence the rest of the paper. The complete list of corrections is below, including typos we noticed. • p. 516, Proposition 2.7: “linear continuous map” is replaced with “linear sequentially continuous map”. • p. 517, Proposition 2.9: “bilinear separately continuous map” is replaced with “bilinear jointly sequentially continuous map”. p. 518, Proposition 2.13: “a continuous linear map” is replaced with “a sequentially continuous linear map”. • p. 519, line -1: “a separately continuous bilinear map” is replaced with “a jointly sequentially continuous bilinear map”. • p. 520, lines 1,2: “continuous” is replaced with “sequentially continuous”. • p. 550, Theorem 8.3 (line 1): “separately continuous bilinear map” is replaced by “jointly sequentially continuous bilinear map”. • p. 552, line 7: “separate continuity” is replaced with “joint sequential continuity”. • p. 552, line 9: “continuity” is replaced with “sequential continuity”. • p. 553, lines 8 and 10: “by continuity” is replaced with “by sequential continuity”. • p. 553, line 14: “separate continuity” is replaced with “joint sequential continuity”. • p. 553, line 15: “property (4)” is replaced by “property (3)”. • p. 553, line 15: “is dense” is replaced by “is sequentially dense”.
UR - http://www.scopus.com/inward/record.url?scp=84978943207&partnerID=8YFLogxK
U2 - https://doi.org/10.1353/ajm.2016.0014
DO - https://doi.org/10.1353/ajm.2016.0014
M3 - تعليقَ / نقاش
SN - 0002-9327
VL - 138
SP - 529
EP - 530
JO - American Journal of Mathematics
JF - American Journal of Mathematics
IS - 2
ER -