1. Preliminaries 5
(f) If v is a basis for V then any x ∈ V can be written as a linear combination
of the form (1.1), with a unique set of scalars α
j
, j =1,...,k. These scalars
(which clearly depend on x) are called the components of x with respect to
the basis v.
(g) The set F
k
is a vector space over F and the set of vectors
e
1
=(1, 0, 0,...,0), e
2
=(0, 1, 0,...,0),..., e
k
=(0, 0, 0,...,1),
is a basis for F
k
. This notation will be used throughout the book, and this
basis will be called the standard basis for F
k
.
We will sometimes write dim V = ∞ when V is infinite-dimensional. How-
ever, this is simply a notational convenience and should not be interpreted in
the sense of ordinal or cardinal numbers (see [7]). In a sense, infinite-dimensional
spaces can vary greatly in their “size”; see Section 3.4 for some further discus-
sion of this.
Definition 1.4
Let V, W be vector spaces over F. The Cartesian product V × W is a vec-
tor space with the following vector space operations. For any α ∈ F and any
(x
j
,y
j
) ∈ V ×W , j =1, 2, let
(x
1
,y
1
)+(x
2
,y
2
)=(x
1
+ x
2
,y
1
+ y
2
),α(x
1
,y
1
)=(αx
1
,αy
1
)
(using the corresponding vector space operations in V and W ).
We next describe a typical construction of vector spaces consisting of func-
tions defined on some underlying set.
Definition 1.5
Let S be a set and let V be a vector space over F. We denote the set of
functions f : S → V by F (S, V ). For any α ∈ F and any f, g ∈ F (S, V ), we
define functions f + g and αf in F(S, V )by
(f + g)(x)=f (x)+g(x), (αf)(x)=αf(x),
for all x ∈ S (using the vector space operations in V ). With these definitions
the set F (S, V ) is a vector space over F.
Many of the vector spaces used in functional analysis are of the above
form. From now on, whenever functions are added or multiplied by a scalar the
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