Nonlinear Control Sy.. - Free

2.3. VECTOR SPACES 37
Proof: First we need to show that the set of linear combinations of elements of S is a
subspace of X. This is straightforward since linear combinations of linear combinations of
elements of S are again linear combinations of the elements of S. Denote this subspace
by N. It is immediate that N contains every element of S and thus N C M. For the
converse notice that M is also a subspace which contains S, and therefore contains all
linear combinations of the elements of S. Thus M C N, and the theorem is proved.
2.3.3 Normed Vector Spaces
As defined so far, vector spaces introduce a very useful algebraic structure by incorporating
the operations of vector addition and scalar multiplication. The limitation with the concept
of vector spaces is that the notion of distance associated with metric spaces has been lost.
To recover this notion, we now introduce the concept of norrned vector space.
Definition 2.8 A normed vector space (or simply a nonmed space) is a pair
consisting of a vector space X and a norm 11 11: X -* IR such that
(z) 11 x lI= 0 if and only if x = 0.
(ii) 1
1 Ax 111 x11 VA ER,dxEX.
(iii) 11x+yj11 X 11 >-0
(X,II - I
thus, the norm of a vector X is nonnegative. Also, by defining property (iii), the triangle
inequality,
I1 x-y11=11 x-z+z-yII511x-z11+11z-y 11 dx,yEX (2.4)
holds. Equation (2.4) shows that every normed linear space may be regarded as a metric
space with distance defined by d(x, y) _II x - y 11.
The following example introduces the most commonly used norms in the Euclidean
space R.