A Soft question about writing one theorem in Introduction.

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!

Answer 1

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.