# Dual Space – The dual of the dual is itself

Good morning, Jakarta. (FYI, I’m on Jakarta now, preparing for our Chinese New Year family celebration :D)

Recently, I found some interesting algebraic properties of a vector space. Some kind of abstract object, however the term is popular among other areas of mathematics. Comments are pleased for correction & other additional information.

First, to make familiar with the ‘muggles’ (#orang awam matematik), let us define some terms:

A field $F (+,\times )$ is a set, such that for each $a,b,c \in F$ the following is hold:

• $a+b = b+a$
• $(a+b)+c = a+(b+c)$
• There exist $0 \in F$ such that $0+a=a+0=a$
• There exist $\ -a \in F$ such that $-a+a=a+(-a)=0$
• $a\times b = b\times a$
• $(a\times b)\times c = a\times (b\times c)$
• There exist $1 \in F$ such that $1\times a=a \times 1=a$
• There exist $a^{-1} \in F-\{0\}$ such that $latex a^{-1} \times a=1$
• $a\times (b+c) = (a\times b) + (a\times c)$
• $(b+c)\times a = (b\times a) + (c\times a)$

A vector space $V$ over a field $F$ is a set, such that for each $u,v,w \in V$ and $a,b \in F$ the following is hold:

• $u+v = v+u$
• $(u+v)+w = u+(v+w)$
• There exist $0 \in V$ such that $0+u=u+0=u$
• There exist $\ -u \in V$ such that $-u+u=u+(-u)=0$
• $a(u+v)=au+av$
• $(a+b)u=au+bu$
• $a(bu)=(ab)u$
• There exist $1 \in F$ such that $1\times v=v \times 1=v$

An isomorphism from a set $A$ to $B$ is a bijection from A to B in which the operations involved in each sets are preserved. Naively, to operate first then transform is the same as to transform first then operate.

A homomorphism from a set $A$ to $B$ is a mapping from A to B in which the operations involved in each sets are preserved. (no need the mapping to be a bijection)

The set $A$ is said to be isomorphic to $B$ if we can find any isomorphism from $A$ to $B$.

Now let the magic begin. Suppose we have a vector space $V$ over a field $F$. Then we can find some homomorphism from $V$ to $F$. Moreover, if we collect every single homomorphisms and gather them in a set, namely $S:=\{\varphi \mid \varphi$ homomorphism from $V$ to $F \}$. Then we can make an operation $+ : S \times S \rightarrow S$ such that $S$ is a vector space under operation $+$. Next, as we always construct a vector space $S$ from arbitrary vector space $V$, we may define $S$ as the dual space of $V$.

Well, we now have the structure of dual space. Now, let we have a vector space $V$ and its dual $V'$. As the $V'$ is also a vector space, we can construct its dual, let say $V''$. Magically, one had proven that $V''$ is then isomorphic to $V$. 2 isomorphic sets means that the two sets are algebraically equivalent. Then, we can conclude that every vector space is equivalent to the dual of its dual.

What a properties..

Moreover, the concept of duality is not only exist in linear algebra. You may check that the duality may appear on linear programming, graph theory, and so on. I believe that the concept of these things are not different. In abstract they are ‘isomorphic’, but somehow I have not discovered how the may come together as one.  That’s all guys. May the magical properties of algebra fulfil your heart. Thanks for your attention. 😀

Enjoy algebra!