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首页英国本科线性泛函分析入门指南
英国本科线性泛函分析入门指南
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线性泛函分析是一本专为英国大学本科课程最后一年学生设计的教材,深入介绍了该领域的核心概念和方法。它建立在读者具备基础的线性代数和实分析知识之上,包括度量空间理论。本书由SpringerUndergraduateMathematicsSeries顾问委员会编著,成员包括来自杜伦大学、牛津大学、伦敦玛丽女王大学、剑桥大学、牛津大学和巴斯大学的学者,如M.A.J.查普曼、K.埃德曼、A.M.麦克因泰尔等。 书中涵盖了广泛的主题,从线性代数的基本概念,如向量空间、线性映射和特征值问题,扩展到更高级的主题,如Banach空间和希尔伯特空间的性质,以及泛函分析中的连续性和收敛性理论。此外,它还讨论了函数空间、算子理论和谱理论等关键概念,这些都是解决实际问题如偏微分方程、计算机图形学和数值计算中的优化问题的基础。 作者们通过实例和应用案例,帮助读者理解这些理论的实际应用,并强调理论与实践的结合。读者将学到如何构造和操作线性泛函,如何理解和利用内积、范数和正交分解,以及如何运用极限和连续性的概念来分析问题。 本书不仅是数学专业学生的宝贵参考资料,也是对数学分析感兴趣的工程师、物理学家和其他科学领域专业人士的实用工具。它提供了一个系统的学习框架,旨在培养学生的抽象思维能力,为他们在后续研究或职业生涯中处理复杂问题奠定坚实的基础。 "linear functional analysis"是线性代数和实分析的延伸,强调了数学分析的实践应用和理论深度,适合那些渴望深入理解并运用线性泛函分析工具的学生和专业人士。
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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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6 Linear Functional Analysis
process will be as in Definition 1.5. We note that the zero element in F (S, V )
is the function which is identically equal to the zero element of V . Also, if
S contains infinitely many elements and V = {0} then F (S, V ) is infinite-
dimensional.
Example 1.6
If S is the set of integers {1,...,k} then the set F (S, F) can be identified with
the space F
k
(by identifying an element x ∈ F
k
with the function f ∈ F (S, F)
defined by f (j)=x
j
,1≤ j ≤ k).
Often, in the construction in Definition 1.5, the set S is a vector space, and
only a subset of the set of all functions f : S → V is considered. In particular,
in this case the most important functions to consider are those which preserve
the linear structure of the vector spaces in the sense of the following definition.
Definition 1.7
Let V , W be vector spaces over the same scalar field F. A function T : V → W
is called a linear transformation (or mapping) if, for all α, β ∈ F and x, y ∈ V ,
T (αx + βy)=αT(x)+βT(y).
The set of all linear transformations T : V → W will be denoted by L(V,W).
With the scalar multiplication and vector addition defined in Definition 1.5 the
set L(V, W) is a vector space (it is a subspace of F (V, W)). When V = W we
abbreviate L(V,V )toL(V ).
A particularly simple linear transformation in L(V ) is defined by I
V
(x)=x,
for x ∈ V .Thisiscalledtheidentity transformation on V (usually we use the
notation I if it is clear what space the transformation is acting on).
Whenever we discuss linear transformations T : V → W it will be taken for
granted, without being explicitly stated, that V and W are vector spaces over
the same scalar field.
Since linear transformations are functions they can be composed (when they
act on appropriate spaces). The following lemmas are immediate consequences
of the definition of a linear transformation.
Lemma 1.8
Let V, W, X be vector spaces and T ∈ L(V,W), S ∈ L(W, X). Then the
composition S
◦
T ∈ L(V, X).
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1. Preliminaries 7
Lemma 1.9
Let V be a vector space, R, S, T ∈ L(V ), and α ∈ F. Then:
(a) R
◦
(S
◦
T )=(R
◦
S)
◦
T ;
(b) R
◦
(S + T )=R
◦
S + R
◦
T ;
(c) (S + T )
◦
R = S
◦
R + T
◦
R;
(d) I
V
◦
T = T
◦
I
V
= T ;
(e) (αS)
◦
T = α(S
◦
T )=S
◦
(αT ).
These properties also hold for linear transformations between different spaces
when the relevant operations make sense (for instance, (a) holds when T ∈
L(V,W), S ∈ L(W, X)andR ∈ L(X, Y ), for vector spaces V, W, X, Y ).
The five properties listed in Lemma 1.9 are exactly the extra axioms which
a vector space must satisfy in order to be an algebra. Since this is the only
example of an algebra which we will meet in this book we will not discuss this
further, but we note that an algebra is both a vector space and a ring, see [5].
When dealing with the composition of linear transformations S, T it is
conventional to omit the symbol
◦
and simply write ST. Eventually we will do
this, but for now we retain the symbol
◦
.
The following lemma gives some further elementary properties of linear
transformations.
Lemma 1.10
Let V, W be vector spaces and T ∈ L(V,W).
(a) T (0) = 0.
(b) If U is a linear subspace of V then the set T (U) is a linear subspace of W
and dim T (U) ≤ dim U (as either finite numbers or ∞).
(c) If U is a linear subspace of W then the set {x ∈ V : T(x) ∈ U } is a linear
subspace of V .
We can now state some standard terminology.
Definition 1.11
Let V, W be vector spaces and T ∈ L(V,W).
(a) The image of T (often known as the range of T ) is the subspace Im T =
T (V ); the rank of T is the number r(T ) = dim(Im T ).
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8 Linear Functional Analysis
(b) The kernel of T (often known as the null-space of T ) is the subspace Ker T =
{x ∈ V : T (x)=0};thenullity of T is the number n(T )=dim(KerT ).
The rank and nullity, r(T ), n(T ), may have the value ∞.
(c) T has finite rank if r(T ) is finite.
(d) T is one-to-one if, for any y ∈ W , the equation T (x)=y hasatmostone
solution x.
(e) T is onto if, for any y ∈ W , the equation T (x)=y has at least one
solution x.
(f) T is bijective if, for any y ∈ W , the equation T (x)=
y has exactly one
solution x (that
is, T is both one-to-one and onto).
Lemma 1.12
Let V, W be vector spaces and T ∈ L(V,W).
(a) T is one-to-one if and only if the equation T (x) = 0 has only the solution
x =0.ThisisequivalenttoKerT = {0} or n(T )=0.
(b) T is onto if and only if Im T = W .IfdimW is finite this is equivalent to
r(T )=dimW .
(c) T ∈ L(V, W) is bijective if and only if there exists a unique transformation
S ∈ L(W, V ) which is bijective and S
◦
T = I
V
and T
◦
S = I
W
.
If V is k-dimensional then
n(T )+r(T )=k
(in particular, r(T ) is necessarily finite, irrespective of whether W is finite-
dimensional). Hence, if W is also k-dimensional then T is bijective if and only
if n(T )=0.
Related to the bijectivity, or otherwise, of a transformation T from a space
to itself we have the following definition, which will be extremely important
later.
Definition 1.13
Let V be a vector space and T ∈ L(V ). A scalar λ ∈ F is an eigenvalue of
T if the equation T(x)=λx has a non-zero solution x ∈ V ,andanysuch
non-zero solution is an eigenvector. The subspace Ker (T − λI) ⊂ V is called
the eigenspace (corresponding to λ) and the multiplicity of λ is the number
m
λ
= n(T − λI).
![](https://csdnimg.cn/release/download_crawler_static/10677637/bg14.jpg)
1. Preliminaries 9
Lemma 1.14
Let V be a vector space and let T ∈ L(V ). Let {λ
1
,...,λ
k
} be a set of distinct
eigenvalues of T , and for each 1 ≤ j ≤ k let x
j
be an eigenvector corresponding
to λ
j
. Then the set {x
1
,...,x
k
} is linearly independent.
Linear transformations between finite-dimensional vector spaces are closely
related to matrices. For any integers m, n ≥ 1, let M
mn
(F) denote the set of all
m ×n matrices with entries in F.AtypicalelementofM
mn
(F) will be written
as [a
ij
] (or [a
ij
]
mn
if it is necessary to emphasize the size of the matrix). Any
matrix C =[c
ij
] ∈ M
mn
(F) induces a linear transformation T
C
∈ L(F
n
, F
m
)as
follows: for any x ∈ F
n
,letT
C
x = y,wherey ∈ F
m
is defined by
y
i
=
n
j=1
c
ij
x
j
, 1 ≤ i ≤ m.
Note that, if we were to regard x and y as column vectors then this trans-
formation corresponds to standard matrix multiplication. However, mainly for
notational purposes, it is generally convenient to regard elements of F
k
as row
vectors. This convention will always be used below, except when we specifically
wish to perform computations of matrices acting on vectors, and then it will
be convenient to use column vector notation.
On the other hand, if U and V are finite-dimensional vector spaces then a
linear transformation T ∈ L(U, V ) can be represented in terms of a matrix. To
fix our notation we briefly review this representation (see Chapter 7 of [1] for
further details). Suppose that U is n-dimensional and V is m-dimensional, with
bases u = {u
1
,...,u
n
} and v = {v
1
,...,v
m
} respectively. Any vector a ∈ U
can be represented in the form
a =
n
j=1
α
j
u
j
,
for a unique collection of scalars α
1
,...,α
n
. We define the column matrix
A =
⎡
⎢
⎣
α
1
.
.
.
α
n
⎤
⎥
⎦
∈ M
n1
(F).
The mapping a → A is a bijective linear transformation from U to M
n1
(F),
that is, there is a one-to-one correspondence between vectors a ∈ U and column
matrices A ∈ M
n1
(F). There is a similar correspondence between vectors b ∈ V
and column matrices B ∈ M
m1
(F). Now, for any 1 ≤ j ≤ n, the vector Tu
j
has the representation
Tu
j
=
m
i=1
τ
ij
v
i
,
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