Page 57 - Advanced Linear Algebra
P. 57
Vector Spaces 41
-
Definition Let < ~¸= 2¹ be any family of vector spaces over . The
direct product of is the vector space
<
H ¢ 2 =~ = d ² ³ = ¦ I
2 2
2
thought of as a subspace of the vector space of all functions from to = .
It will prove more useful to restrict the set of functions to those with finite
support.
Definition Let < ~¸= 2¹ be a family of vector spaces over - . The
support of a function ¢ 2 ¦ = is the set
supp² ³~¸ 2 ² ³£ ¹
Thus, a function has finite support if ² ³ ~ for all but a finite number of
2. The external direct sum of the family is the vector space
<
ext
¦
=~ H ¢ 2 = ² ³ = , has finite support I
d
2 2
thought of as a subspace of the vector space of all functions from to = .
2
An important special case occurs when =~ = for all 2 . If we let = 2
denote the set of all functions from 2 to and ² = 2 ³ denote the set of all
=
functions in = 2 that have finite support, then
ext
2
2
=~ = and =~ ²= ³
2 2
Note that the direct product and the external direct sum are the same for a finite
family of vector spaces.
Internal Direct Sums
An internal version of the direct sum construction is often more relevant.
)
Definition A vector space = is the (internal direct sum of a family
< ~¸: 0¹ of subspaces of , written
=
=~ or =~ < :
0
if the following hold:
