Generalizing dual spaces- A study on functionals.

by ayushkhaitan3437

A functional is that which maps a vector space to a scalar field like \Bbb{R} or \Bbb{C}. If X is the vector space under consideration, and f_i:X\to \Bbb{R} (or f_i:X\to\Bbb{C}), then the vector space \{f_i\} of functionals is referred to as the algebraic dual space X^*. Similarly, the vector space of functionals f'_i:X^*\to \Bbb{R} (or f'_i:X^*\to\Bbb{C}) is referred to as the second algebraic dual space. It is also referred to as X^{**}.

How should one imagine X^*? Imagine a bunch of functionals being mapped to \Bbb{R}. One way to do it is to make all of them map only one particular x\in X. Hence, g_x:X^*\to \Bbb{R} such that g_x(f)=g(f(x)). Another such mapping is g_y. The vector space X^{**} is isomorphic to X.

My book only talks about X, X^* and X^{**}. I shall talk about X^{***}, X^{****}, and X^{**\dots *}. Generalization does indeed help the mind figure out the complete picture.

Say we have X^{n*} (n asterisks). Imagine a mapping X^{n*}\to \Bbb{R}. Under what conditions is this mapping well-defined? When we have only one image for each element of X^{n*}. Notice that each mapping f:X^{n*}\to \Bbb{R} is an element of the vector space X^{(n+1)*}. To make f a well-defined mapping, we select any one element a\in X^{(n-1)*}, and determine the value of each element of X^{n*} at a. One must note here that a is a mapping (a: X^{(n-2)*}\to\Bbb{R}). What element in X^{(n-2)*} that a must map to \Bbb{R} should be mentioned in advance. Similarly, every element in X^{(n-2)*} is also a mapping, and what element it should map from X^{(n-3)*} should also be pre-stated.

Hence, for every element in X^{n*}, one element each from X^{(n-2)*}, X^{(n-3)*},X^{(n-4)*},\dots ,X should be pre-stated. For every such element in X^{n*}, this (n-2)-tuple can be different. To define a well-defined mapping f:X^{n*}\to \Bbb{R}, we choose one particular element b\in X^{(n-1)*}, and call the mapping f_b. Hence,

f_b(X^{n*})=X^{n*}(b, rest of the  (n-2)-tuple ),

f_c(X^{n*})=X^{n*}(c, rest of the (n-2)-tuple), and so on.

 

By

f_b(X^{n*})=X^{n*}(b, rest of the  (n-2)-tuple),

we mean the value of every element of X^{n*} at (b, rest of the (n-2)-tuple).

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