Introductory Functional Analysis with Applications
byErwin Kreyszig
x = (ξ₁, ξ₂, …) briefly x = (ξᵢ)
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metric space (X, d) briefly X
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J: the collection of all open subsets of X
topological space (X, J)
The set J is called a topology for X.
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A metric space is a topological space
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Let M be a subset of a metric space X. Then a point x0∈X (which mayor may not be a point of M) is called an 【accumulation point(or limit point)】of M if every neighborhood of x0 contains at least one point y∈M distinct from x0.
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The set consisting of the points of M and the accumulation points of M is called the 【closure】 of M
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A subset M of a metric space X is said to be 【dense】 in X if closure(M)=X
separable = countable + dense
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completeness is an additional property which a metric space may or may not have
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Cauchy sequence ： if a sequence (xₙ) satisfies the condition of the Cauchy criterion,
we may call it a ～
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the concept of convergence is not an intrinsic property of the sequence itself but also depends on the space in which the sequence lies.
In other ·words, a convergent sequence is not convergent "on its own" but it must converge to some point in the space.
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Completeness of the real line R is also the main reason why in calculus we use R rather than the rational line Q
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之所以提出Completeness，是为了避免类似如下的囧境：
- 在有理数域上做微积分运算，所得的计算结果不是有理数；
- 在metric space (X, d)上做运算，所得的计算结果不在 X里
为了解决这样的囧境，把计算结果也添加到X里，从而形成一个新的集合X'。不再以X为研究对象，而以X'为研究对象。
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l2： Hilbert sequence space
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CH2
normed space (X, ||.||)= vector space + metric defined by norm
Banach space = normed space + complete
operator: A mapping from a normed space X into a normed space Y
functional: A mapping from X into the scalar field R or C
linear operator is continuous IFF it is bounded.
algebraic dual space X*： all bounded linear functionals on vector space X
θ：zero vector
Quotient space, codimension，Fig. 12,13, 14
Can every incomplete normed space be completed? YES
2.3-1 Theorem (Subspace of a Banach space).
A subspace Y of a Banach space X is complete if and only if the set Y is ciosed in X.
If a normed space X has a Schauder basis, then X is separable
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2.3-2 Theorem (Completion).
2.4-2 Theorem (Completeness).
Every finite dimensional subspace Y of a normed space X is complete.
In particular, every finite dimensional normed space is complete.
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2.4-3 Theorem (Closedness).
Every finite dimensional subspace Y of a normel1 space X is closed in X.
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Note that infinite dimensional subspaces need not be closed
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Equivalent norms on X define the same topology for X.
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如何通俗地理解open set， close set？
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2.4-5 Theorem (Equivalent norms).
On a finite dimensional vector space X, any norm ||.|| is equivalent to any other norm ||.||0
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通俗理解，为比较vector space X里元素的大小，选择||.||, ||.||2, ||.||3, ||.||p中的任何一个，元素按大小排列出来的顺序是一致的。
”元素按大小排列出来的顺序“就是书里说的”拓扑”
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CH 5
prior estimate can be used at the beginning of a calculation for estimating the number of steps necessary to obtain a given accuracy.
posterior estimate can be used at intermediate stages or at the end of a calculation. It is at least as accurate as (5) and may be better;
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5.2-2 Jacobi iteration
This shows that, roughly speaking, convergence is guaranteed if the
elements in the principal diagonal of A are sufficiently large.
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Compact sets are important since they are "well-behaved": they
have several basic properties similar to those of finite sets and not
shared by noncompact sets
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Another word for null space is "kernel."
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The inverse of a linear operator exists if and only if the null space of the operator consists of the zero vector only.(Tx=0 ==> x=0)
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identity operator I: Ix = x
zero operator 0: 0x = 0
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If dim D(T)=n<∞, then dim R(T)<=n
这个性质有点意思。影射后的空间的XX不会大于影射前的空间的XX。（但如何更直观地理解XX表示的是什么？比如，把XX理解为信息，数据量）
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2.7-8 Theorem (Finite dimension). If a normed space X is finite
dimensional, then every linear operator on X is bounded.
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If T is continuous at a single point ---> T is continuous at all point(传染性) ←--> T is bounded.
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如何通俗地理解operator norm ||T||？它是用来衡量什么的？
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the set of all linear functionals defined on a vector space X can itself be made into a vector space.
This space is denoted by X* and is called the algebraic dual space of X.
(f1 + f2)(x) = f1(x) + f2(x)
(af)(x) = af(x)
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对于f(x), 以前f是固定的,x是变化的；现在x是固定的，f是变化的。
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X*: the second 【algebraic dual space】 of X.
(X*)*=X**
C:X--->X**, C is isomorphic
C is also called the canonical embedding of X into X**. (X⊂X**)
若X=X**，X is algebraically 【reflexive】
（if X is finite dimensional, then X is algebraically reflexive.）
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whenever the structure is the primary object of study, whereas the nature of the points does not matter. This situation occurs quite often. It suggests the concept of an isomorphism.
By definition, this is a bijective mapping of X onto X' which preserves the structure.
.d'(Tx, Ty)= d(x, y)
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Linear operators on finite dimensional vector spaces can be rep-
resented in terms of matrices, as explained below. In this way, matrices
become the most important tools for studying linear operators in the
finite dimensional case.
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The dual space of lp is lq; here, 1<p<∞ and q is the conjugate of p, that is, l/p + 1/q = 1.
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CH3
inner product space (or pre-Hilbert space): vector space+ <,>
||x||=sqrt(<x,x>), d(x, y) = sqrt(<x-y, x-y>)
Hilbert space： inner product space + complete
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metric和norm的相互转换：
metric to norm:
||x|| = ||d(x,0)||
norm to metric:
Re<x, y> = (||x+y||^2 - ||x-y||^2)/4 polarizatitJn identity
Im<x, y> = (||x+iy||^2 - ||x-iy||^2)/4 polarizatitJn identity
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Schwarz inequality: |<x,y>| <=||x|| ||y||
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An isomorphism T of an inner product space X:
<Tx, Ty> = <x, y>
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Problem 3.7 ≌x⊥y iff ||x+ay|| = ||x -ay|| (放射坐标系？)
||T||描述的是(T从domain到range的)最大变化率。类似T
= max f'(x)
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3.3-3 Definition (Direct sum).
A vector space X is said to be the 【direct sum】 of two subs paces Y and Z of X, written
X= Y⊕Z,
if each x∈X has a unique representation x=y+Z , y∈Y,z∈Z.
Then Z is called an 【algebraic complement】 of Y in X and vice versa,
and Y, Z is called a 【complementary pair】 of subspaces in X.
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general Hilbert space H, closed subspace Y of H
H = Y ⊕ Y⊥, Y⊥= {z∈H |z⊥Y}
P: H-->Y. P is called (orthogonal) projection (or projection operator)
P: Y-->Y
P: Y⊥ -->0
PP = P
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orthogonality的意义：
给定坐标基ei和坐标系下的一个向量x，可以求得x在ei上的坐标：
<x, e0> = a0, …..
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total set (or fundamental set)
P168
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