CHAPTER 1
Normed Linear Spaces
A linear space is simply an abstract version of the familiar vector spaces R, R
2
,
R
3
and so on. Recall that vector spaces have certain algebraic properties: vectors
may be added, multiplied by scalars, and vector spaces have bases and subspaces.
Linear maps between vector spaces may be described in terms of matrices. Using the
Euclidean norm or distance, vector spaces have other analytic properties (though
you may not have called them that): for example, certain functions from R to R
are continuous, differentiable, Riemann integrable and so on.
We need to make three steps of generalization.
Bases: The first is familiar: instead of, for example, R
3
, we shall sometimes want
to talk about an abstract three–dimensional vector space V over the field R. This
distinction amounts to having a specific basis {e
1
, e
2
, e
3
} in mind, in which case
every element of V corresponds to a triple (a, b, c) = ae
1
+ be
2
+ ce
3
of reals – or
choosing not to think of a specific basis, in which case the elements of V are just
abstract vectors v. In the abstract language we talk about linear maps or operators
between vector spaces; after choosing a basis linear maps become matrices – though
in an infinite dimensional setting it is rarely useful to think in terms of matrices.
Ground fields: The second is fairly trivial and is also familiar: the ground field
can be any field. We shall only be interested in R (real vector spaces) and C
(complex vector spaces). Notice that C is itself a two–dimensional vector space
over R with additional structure (multiplication). Choosing a basis {1, i} for C
over R we may identify z ∈ C with the vector (<(z), =(z)) ∈ R
2
.
Dimension: In linear algebra courses, you deal with finite dimensional vector
spaces. Such spaces (over a fixed ground field) are determined up to isomor-
phism by their dimension. We shall be mainly looking at linear spaces that are
not finite–dimensional, and several new features appear. All of these features may
be summed up in one line: the algebra of infinite dimensional linear spaces is in-
timately connected to the topology. For example, linear maps between R
2
and R
2
are automatically continuous. For infinite dimensional spaces, some linear maps
are not continuous.
1. Linear (vector) spaces
Definition 1.1. A linear space over a field k is a set V equipped with maps
⊕ : V × V → V and · : k × V → V with the properties
(1) x ⊕ y = y ⊕x for all x, y ∈ V (addition is commutative);
(2) (x ⊕ y) ⊕z = x ⊕ (y ⊕z) for all x, y, z ∈ V (addition is associative);
(3) there is an element 0 ∈ V such that x ⊕ 0 = 0 ⊕ x = x for all x ∈ V (a zero
element);
(4) for each x ∈ V there is a unique element −x ∈ V with x ⊕ (−x) = 0 (additive
inverses);
5
我的内容管理
收起
我的收益 登录查看自己的收益
我的积分
登录查看自己的积分
我的C币
登录后查看C币余额
我的收藏 
我的下载 
下载帮助 
评论0