Understanding dual spaces

Question

My professor asked me to give him an example about dual spaces from our real life so I was thinking if we take the set of all our actions as the space X can the space of all our thoughts be the dual space ?!!

It is clear that the set of actions is a vector space : Adding two action give an action and multiplying an action by a scalar give an action ( the addition means doing more than one action and multiplication by a scalar means increase or decrease the power of the action). The norm of this space is the time we need to do an action. So if time separating a sequence of actions is very small we say that this sequence converges The reason I thing that the dual of this space should be the space of thought is that thoughts and actions are connected to each other and if we have a convergente sequence of actions we surely have a convergente sequence of thoughts but the inverse isn't true we can have a convergente sequence of thought without actions.

 What do you think about this example, should I explain it to my professor ??????

Answer 1

A concrete example is the force applying on an object. The newton second law of motion said that Force = mass x acceleration. It is clear that here the mathematical representation of force is functional that map the set of masses to a real number. Then, if we take the set of masses as a space X. Then it dual would be the set of forces.

Answer 2

A possible example is the following. We have a shop with a (finite) set of articles $S$. Let $V$ be the vector space of prices for the objects in $S$, then a choice of things to buy would give you an element of the dual space $V^*$ (seen as a functional giving you back how much do you spend depending on the price tags).