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.
By
, rest of the (n-2)-tuple
,
we mean the value of every element of at
rest of the (n-2)-tuple
.