Generalizing dual spaces- A study on functionals.
A functional is that which maps a vector space to a scalar field like or . If is the vector space under consideration, and (or ), then the vector space of functionals is referred to as the algebraic dual space . Similarly, the vector space of functionals (or ) is referred to as the second algebraic dual space. It is also referred to as .
How should one imagine ? Imagine a bunch of functionals being mapped to . One way to do it is to make all of them map only one particular . Hence, such that . Another such mapping is . The vector space is isomorphic to .
My book only talks about and . I shall talk about , and . Generalization does indeed help the mind figure out the complete picture.
Say we have ( asterisks). Imagine a mapping . Under what conditions is this mapping well-defined? When we have only one image for each element of . Notice that each mapping is an element of the vector space . To make a well-defined mapping, we select any one element , and determine the value of each element of at . One must note here that is a mapping (). What element in that must map to should be mentioned in advance. Similarly, every element in is also a mapping, and what element it should map from should also be pre-stated.
Hence, for every element in , one element each from should be pre-stated. For every such element in , this -tuple can be different. To define a well-defined mapping , we choose one particular element , and call the mapping . Hence,
rest of the (n-2)-tuple ,
rest of the (n-2)-tuple, and so on.
, rest of the (n-2)-tuple,
we mean the value of every element of at rest of the (n-2)-tuple.