Analysis for Applications
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1.5 Vector spaces
A vector space over an algebraic eld K is a set X together with two
algebraic operations: vector addition and scalar multiplication. Elements
x 2 X are called vectors; elements 2 K are called scalars. For our
purposes, the eld K will always be either IR or C. Vector addition is a
function, denoted \+" which maps X X into X and satises for every
x; y; and z in X :
1. x + y = y + x (commutativity),
2. x + (y + z) = (x + y) + z (associativity).
Furthermore, there exists a vector in X , denoted 0, and for every vector
x 2 X there exists a vector ,x such that
x + 0 = x; and x + (,x) = 0:
Scalar multiplication is a function mapping K X into X , which satises
for all x; y 2 X and all ; 2 K ,
1. (x) = ( )x,
2. 1x = x (1 denotes the multiplicative identity in K ),
3. (x + y) = x + y,
4. ( + )x = x + x.
A vector space dened with respect to real scalars (that is, K = IR) is
called a real vector space; a vector space dened with respect to complex
scalars is called a complex vector space.
Examples (check that these satisfy the denitions)
IR and C are themselves vector spaces, as are IRn and Cn with the
usual denitions of vector addition and scalar multiplication.
The spaces l , 1 p 1 are vector spaces with vector addition
dened by
x + y = (x1 + y1 ; x2 + y2 ; : : :) for x = (x ); y = (y )
and scalar multiplication dened by
x = (x1 ; x2 ; : : :):
p
j
j
D. Dobson
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C [a; b] is a vector space with
(f + g)(x) = f (x) + g(x) for x 2 [a; b];
and
(f )(x) = f (x) for x 2 [a; b]
for any f; g 2 C [a; b].
It is important to note that both vector addition and scalar multiplication
must dene values in the vector space, that is, x 2 X and x + y 2 X for
all scalars and all x; y 2 X .
You may recall that in metric spaces, we dened a subspace to be any
subset of elements of the original space, paired with the induced metric.
In vector spaces, we will not refer to arbitrary subsets as subspaces. To
be a called a subspace, we require that the set is \algebraically closed"
under the induced operations of vector addition and scalar multiplication.
More denitions. A subspace of a vector space X is a subset Y X such
that for all y1; y2 2 Y , and all scalars we have y1 2 Y and y1 + y2 2 Y .
For any subset M X , the set of all (nite!) linear combinations of
vectors in M
1 x1 + 2 x2 + + x
is called the span of M , written span M . Note that span M is a subspace
of X .
The nite set of vectors x1; x2 ; : : : ; x 2 X is called linearly independent
n
if the only solution to the equation
n
n
1 x1 + 2 x2 + + x = 0
n
n
is 1 = 2 = = = 0. If the vectors x1 ; x2 ; : : : ; x are not linearly
independent, they are called linearly dependent. An arbitrary subset of
vectors M X is called linearly independent if every nonempty nite
subset of M is linearly independent. M is called linearly dependent if it
is not linearly independent.
A vector space X is said to be nite dimensional if there exists a
natural number N such that X contains a linearly independent set of N
vectors, but any set of N + 1 or more vectors in X is linearly dependent.
N is called the dimension of X . If X is not nite dimensional, it is called
innite dimensional.
n
n
Analysis for Applications
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Examples. IRn and Cn are nite dimensional, with dimension n. l and
p
C [a; b] are innite dimensional.
Denition. A linearly independent subset
called a Hamel basis for X .
B
X which spans X is
Theorem 1.5.1 Every nonempty vector space X has a Hamel basis.
Proof. A proof can be found in most functional analysis texts but is
outside the scope of our discussion. The innite dimensional case requires
Zorn's lemma.
Note: If B is a basis, then every element x 2 X has a unique representation
as a (nite) linear combination of elements of B .
1.6 Banach spaces
A normed vector space is a vector space together with a norm. A norm
on a vector space X is a real-valued function dened on X , whose value
at x 2 X is denoted kxk and which satises
1. kxk 0,
2. kxk = 0 if and only if x = 0,
3. kxk = jjkxk for all scalars ,
4. kx + yk kxk + kyk.
Note that a norm on X denes the metric
d(x; y ) = kx , y k;
called the metric induced by the norm. You should check for yourself that
this is always a metric. Other metrics can of course be dened in terms
of the norm, but they are generally less important in what follows.
Denition. A Banach space is a normed vector space which is complete
in the metric induced by the norm.
D. Dobson
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Examples.
IRn and C are Banach spaces with the norm as we dened in our
n
discussion of linear algebra.
We have already dened the following spaces as metric spaces. To show
that they are Banach spaces requires that we rst check that the norm
as dened is indeed a norm, and second that the space is complete in the
metric induced by the norm. We will study closely related spaces in more
detail soon, so for now we put o the details.
The spaces l , 1 p 1 are Banach spaces with the norms
p
01
11
X
kxk = @ jx j A ; p < 1
=p
p
p
j
j
and
=1
kxk1 = sup jx j; p = 1:
j
j
The space C [a; b] is a Banach space with the norm
kf k1 = max
2[ ] jf (t)j:
t
a;b
1.7 Compactness
A metric space X is compact if every sequence in X has a convergent
subsequence. A subset M X is called compact if every sequence in M
has a convergent subsequence whose limit is in M .
It should be noted that compactness as we have dened it is known
as sequential compactness in general topological spaces. In metric spaces
this denition is equivalent to other standard notions of compactness.
It is usually proved (or dened) in advanced calculus courses that subsets of IRn are compact if and only if they are closed and bounded. The
same fact holds in normed vector spaces provided the space is nite dimensional:
Theorem 1.7.1 If X is a nite dimensional normed vector space then
each subset M X is compact if and only if M is closed and bounded.