A Soft question about writing one theorem in Introduction.

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I am doing a program and I want to write the theorem below in my introduction briefly and clearly. But it seems that I have to write the whole theorem in my introduction. Is there a better way to describe the theorem below.



Theorem: Let $X$ be a normed space. Then the following statements are equivalent.



(a) $X$ is reflexive.



(b) $sigma(X^*,X)=sigma(X^*,X^**)$.



(c) ball $X$ is weakly compact in $X$.



Furthermore, each of $(a)$-$(c)$ implies the following



(d) $X^*$ is reflexive,



and $(a)$-$(d)$ are equivalent if $X$ is a Banach space.



Thank you in advance!






functional-analysis soft-question banach-spaces normed-spaces







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  • (b) What is $sigma$? (c) Isn't it the unit ball?
    – J.-E. Pin
    Aug 31 at 3:36














up vote
0
down vote

favorite












I am doing a program and I want to write the theorem below in my introduction briefly and clearly. But it seems that I have to write the whole theorem in my introduction. Is there a better way to describe the theorem below.



Theorem: Let $X$ be a normed space. Then the following statements are equivalent.



(a) $X$ is reflexive.



(b) $sigma(X^*,X)=sigma(X^*,X^**)$.



(c) ball $X$ is weakly compact in $X$.



Furthermore, each of $(a)$-$(c)$ implies the following



(d) $X^*$ is reflexive,



and $(a)$-$(d)$ are equivalent if $X$ is a Banach space.



Thank you in advance!






functional-analysis soft-question banach-spaces normed-spaces







share|cite|improve this question





















  • (b) What is $sigma$? (c) Isn't it the unit ball?
    – J.-E. Pin
    Aug 31 at 3:36












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I am doing a program and I want to write the theorem below in my introduction briefly and clearly. But it seems that I have to write the whole theorem in my introduction. Is there a better way to describe the theorem below.



Theorem: Let $X$ be a normed space. Then the following statements are equivalent.



(a) $X$ is reflexive.



(b) $sigma(X^*,X)=sigma(X^*,X^**)$.



(c) ball $X$ is weakly compact in $X$.



Furthermore, each of $(a)$-$(c)$ implies the following



(d) $X^*$ is reflexive,



and $(a)$-$(d)$ are equivalent if $X$ is a Banach space.



Thank you in advance!






functional-analysis soft-question banach-spaces normed-spaces







share|cite|improve this question













I am doing a program and I want to write the theorem below in my introduction briefly and clearly. But it seems that I have to write the whole theorem in my introduction. Is there a better way to describe the theorem below.



Theorem: Let $X$ be a normed space. Then the following statements are equivalent.



(a) $X$ is reflexive.



(b) $sigma(X^*,X)=sigma(X^*,X^**)$.



(c) ball $X$ is weakly compact in $X$.



Furthermore, each of $(a)$-$(c)$ implies the following



(d) $X^*$ is reflexive,



and $(a)$-$(d)$ are equivalent if $X$ is a Banach space.



Thank you in advance!






functional-analysis soft-question banach-spaces normed-spaces




functional-analysis soft-question banach-spaces normed-spaces






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asked Aug 30 at 2:36









Answer Lee

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  • (b) What is $sigma$? (c) Isn't it the unit ball?
    – J.-E. Pin
    Aug 31 at 3:36
















  • (b) What is $sigma$? (c) Isn't it the unit ball?
    – J.-E. Pin
    Aug 31 at 3:36















(b) What is $sigma$? (c) Isn't it the unit ball?
– J.-E. Pin
Aug 31 at 3:36




(b) What is $sigma$? (c) Isn't it the unit ball?
– J.-E. Pin
Aug 31 at 3:36










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Rule of a thumb: in the introduction, avoid mathematical symbols if possible.
You could write for instance, avoiding any symbol,




We prove the following results:



  1. A normed space is reflexive if and only if its unit ball is weakly compact.

  2. The dual space of a reflexive normed space is reflexive.

  3. A Banach space is reflexive if and only if its dual space is reflexive.






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    Rule of a thumb: in the introduction, avoid mathematical symbols if possible.
    You could write for instance, avoiding any symbol,




    We prove the following results:



    1. A normed space is reflexive if and only if its unit ball is weakly compact.

    2. The dual space of a reflexive normed space is reflexive.

    3. A Banach space is reflexive if and only if its dual space is reflexive.






    share|cite|improve this answer
























      up vote
      0
      down vote













      Rule of a thumb: in the introduction, avoid mathematical symbols if possible.
      You could write for instance, avoiding any symbol,




      We prove the following results:



      1. A normed space is reflexive if and only if its unit ball is weakly compact.

      2. The dual space of a reflexive normed space is reflexive.

      3. A Banach space is reflexive if and only if its dual space is reflexive.






      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote









        Rule of a thumb: in the introduction, avoid mathematical symbols if possible.
        You could write for instance, avoiding any symbol,




        We prove the following results:



        1. A normed space is reflexive if and only if its unit ball is weakly compact.

        2. The dual space of a reflexive normed space is reflexive.

        3. A Banach space is reflexive if and only if its dual space is reflexive.






        share|cite|improve this answer












        Rule of a thumb: in the introduction, avoid mathematical symbols if possible.
        You could write for instance, avoiding any symbol,




        We prove the following results:



        1. A normed space is reflexive if and only if its unit ball is weakly compact.

        2. The dual space of a reflexive normed space is reflexive.

        3. A Banach space is reflexive if and only if its dual space is reflexive.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 31 at 3:42









        J.-E. Pin

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