What is the interest of duality in algebra, and in general in mathematics?

Question

Before to ask my question I precise I'm a chemist, I ask this question because it makes me crazy to don't understand something I learnt in school.

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So I had two years ago a small chapter about duality in algebra, and I have never understood the interest of that, and i think all of my classmate too. But (I don't know why) I'm thinking about it for some month, and i'm still lost.

More, in my home I have this fabulus book writen by Xavier Merlin to help candidat of Polytechnique competitive exam. Unfortunately I'm not in Polytechnique. And even I will "never" use this concept in Chemistry I would like someone explain me why this concept is important in algebra and maybe in all mathematics, and if possible give me an easy example which traduce this importance.

I really should be very glad ! :D

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NB : The autor says in the beginning of the chapter about duality

We are not sure all persons who have this book (we salute) will have the bravery to read this chapter, due to the aversion they have for this strange phenomenon called duality. It's a pity [...]. Especially as nobody understand it well and in the kingdom of the blind, the one-eyed man is king.

Answer 1

My first approach with duality was in projective geometry, where it it is a very powerful tool: we can proof a theorem for points and we have a dual theorem for straight lines !

But duality is a powerful tool in many fields of math applications. Also in chemistry I suppose are used involutions that are an example of duality.

There are really many different case of duality and a first approach can be the use of duality in logic and set theory.

Answer 2

It is very often the case that on ehas a vector space $V$ over some field $k$ and one is interested in linear maps $V\to k$. What is more natural that to collect them into a set? Every time we have some objects we are interested in the first thing we do is it collect them in a set. In this case, we write $V^*$ the set of all linear maps $V\to k$.

Now linear maps $V\to k$ can be added and multiplied by scalars. These operations are very natural and it turns out that endowed with them $V^*$ is another vector space.

So $V^*$ is an object that show up naturally when one is interested in linear forms on a vector space $V$, and we are lucky enough that it is itself a vector space.

Duality is the study of this vector space $V^*$ in its relation to the original one $V$.