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In recent forty years, the development of theoretical study on Banach space is rapid,on the contrary, the theoretical study on locally convex spaces, which is widely spreadin Banach space, develops slowly. The scholars who study on the Drop property, Eke-land variational principle, the Asplund property and Di?erentiation Theory achieve greatsuccesses. Since the beginning of this century, the researches of convexity and smooth-ness of local convex spaces have a rapid development. In this paper, the author makes adis- cussion on the characteristics, properties and the relations of several convexities andsmoothness in local convex spaces. This paper is divided into five chapters:Let X be a real linear space, P a family of separated seminorms on X , (X, P ) adual , (X , TP ) a locally convex space generated by P .Chapter 1 : The knowledge of preparation.Chapter 2 : We give the notions of average strong converxity and average strongsmoothness for dual (X , P ) , and obtained some necessary and su?cient conditionsof average strong converxity (average strong smoothness) in locally convex spaces . Inaddition ,On the conditions of P ? re?exivity,we obtained the dual theorem of betweenaverage strong converxity and average strong smoothness also.Chapter 3 : We give the notions of average weakly local uniform converxity andaverage weakly local uniform smoothness for dual (X , P ) , and obtained some necessaryand su?cient conditions of average weakly local uniform converxity (average weakly localuniform smoothness) in locally convex spaces . In addition ,On the conditions of P ? re-?exivity,we obtained the dual theorem of between average weakly local uniform converxityand average weakly local uniform smoothness also.Chapter 4 : We give the notions of average uniform extreme converxityand averageuniform extreme smoothness for dual (X , P ) , and obtained some necessary and su?cientconditions of average uniform extreme converxity (average uniform extreme smoothness) in locally convex spaces . In addition ,we obtained the dual theorem of between averageuniform extreme converxity and average uniform extreme smoothness also.Chapter 5 : In this section we mainly discusse the relations between convexity andsmooothness in locally convex spaces.
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