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!

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