Page 372 - Advanced Linear Algebra
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356 Advanced Linear Algebra
(
Let us think of the “information” about contained in a function ¢ ( ¦ ) as
the way in which distinguishes elements of using labels from ) . The
(
relationship above implies that
² ³ £ ² ³ ¬ ² ³ £ ² ³
and this can be phrased by saying that whatever ability has to distinguish
elements of is also possessed by . Put another way, except for labeling
(
differences, any information about that is contained in is also contained in
(
.
If happens to be injective, then the only difference between and is the
values of the labels. That is, the two functions have the same information about
( . However, in general, is not required to be injective and so may contain
more information than .
Now consider a family of sets and a family
I
< I ~ ¸ ¢ ( ¦ ? ? ¹
Assume that : I and ¢ ( ¦ : < . If the diagram in Figure 14.1 commutes
for all < , then the information contained in every function in < is also
<
contained in . Moreover, since , the function cannot contain more
information than is contained in the entire family and so we conclude that
contains exactly the same information as is contained in the entire family . In
<
this sense, ¢ ( ¦ : is universal among all functions ¢ ( ¦ ? in .
<
In this way, a single function ¢ ( ¦ : , or more precisely, a single pair ²:Á ³ ,
can capture a mathematical concept as described by a family of functions. Some
examples from linear algebra are basis for a vector space, quotient space, direct
sum and bilinearity (as we will see).
Let us make a formal definition.
Definition Referring to Figure 14.2, let be a set and let be a family of sets.
(
I
Let
< I ~ ¸ ¢ ( ¦ ? ? ¹
be a family of functions, all of which have domain and range a member of .
I
(
Let
> ~ ¸ ¢? ¦ @ ?Á@ ¹
I
be a family of functions with domain and range in . We assume that has the
I
>
following structure:
1 > ) contains the identity function for each member of .
I
:
