Compactness and compact operator——紧性与紧算子

2013-03-12 11:00阅读：

http://blog.sina.cn/dpool/blog/u/2284290302

1.紧集
定义
紧集是拓扑空间内的一类特殊点集，它们的任何开覆盖都有有限子覆盖。在度量空间内，紧集还可以定义为满足以下任一条件的集合：
i)任意列有收敛子列且该子列的极限点属于该集合（自列紧集）；
ii)具备Bolzano-Weierstrass性质；
iii)完备且完全有界。

性质

紧集具有以下性质：
1.点集是紧集的充分必要条件是它为有界闭集。
2.紧集在连续函数下的像仍是紧集。
3.豪斯多夫空间的紧子集是闭集。
4.实数空间的非空紧子集有最大元素和最小元素。
5.Heine-Borel定理:在Rn内，一个集合是紧集当且仅当它是闭集并且有界。
6.定义在紧集上的连续实值函数有界且有最大值和最小值。
7.定义在紧集上的连续实值函数一致连续。

直观理解

从某种意义上，紧集类似于有限集。举最简单的例子而言，在度量空间中，所有的有限集都有最大与最小元素。一般而言，无限集可能不存在最大或最小元素（比如R中的(0, 1)）,但R中的非空紧子集都有最大和最小元素。在很多情况下，对有限集成立的证明可以扩展到紧集。一个简单的例子是对以下性质的证明：定义在紧集上的连续实值函数一致连续。

类似概念

自列紧集：每个有界序列都有收敛的子序列。
可数紧集：每个可数的开覆盖都有一个有限的子覆盖。
伪紧：所有的实值连续函数都是有界的。
弱可数紧致：每个无穷子集都有极限点。
在度量空间中，以上概念均等价于紧集。
以下概念通常弱于紧集：
相对紧致：如果一个子空间Y在母空间X中的闭包是紧致的，则称Y是相对紧致于X。
准紧集：若空间X的子空间Y中的所有序列都有一个收敛的子序列，则称Y是X中的准紧集。

局部紧致空间：如果空间中的每个点都有个由紧致邻域组成的局部基，则称这个空间是局部紧致空间。
参考来源：http://baike.baidu.com/view/1224337.htm
2 相对紧（relatively compact ）
In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact.
Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Such a subset may also be called relatively bounded, or pre-compact, although the latter term is also used for a totally bounded subset. (These are equivalent in a complete space.)
Some major theorems characterise relatively compact subsets, in particular in function spaces. An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterises relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices).
The definition of an almost periodic function F at a conceptual level has to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
As a counterexample take any neighbourhood of the particular point of an infinite particular point space. The neighbourhood itself may be compact but is not relatively compact because its closure is the whole non-compact space.
from：http://en.wikipedia.org/wiki/Relatively_compact
3 紧算子compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous.
Any bounded operator L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting. When Y is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure in the operator norm of the finite-rank operators. Whether this was true in general for Banach spaces (the approximation property) was an unsolved question for many years; in the end Per Enflo gave a counter-example.
The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K on function spaces; the compactness property is shown by equicontinuity. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived from this connection.

Contents

[show]

Equivalent formulations

A bounded operator T is compact if and only if any of the following is true

Image of the unit ball in X under T is relatively compact in Y.
Image of any bounded set under T is relatively compact in Y.
Image of any bounded set under T is totally bounded in Y.
there exists a neighbourhood of 0, $U\subset X$ , and compact set $V\subset Y$ such that $T(U)\subset V$ .
For any sequence $(x_n)_{n\in \mathbb N}$ from the unit ball in X, the sequence $(Tx_n)_{n\in\mathbb N}$ contains a Cauchy subsequence.

Note that if a linear operator is compact, then it is easy to see that it is bounded, and hence continuous.

Important properties

In the following, X, Y, Z, W are Banach spaces, B(X, Y) is the space of bounded operators from X to Y with the operator norm, K(X, Y) is the space of compact operators from X to Y, B(X) = B(X, X), K(X) = K(X, X), $id_X$ is the identity operator on X.

K(X, Y) is a closed subspace of B(X, Y): Let T_n, n ∈ N, be a sequence of compact operators from one Banach space to the other, and suppose that T_n converges to T with respect to the operator norm. Then T is also compact.

$B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z).$ In particular, K(X) forms a two-sided operator ideal in B(X).

$id_X$ is compact if and only if X has finite dimension.

For any T ∈ K(X), $id_X - T$ is a Fredholm operator of index 0. In particular, $\operatorname{im}\,(id_X - T)$ is closed. This is essential in developing the spectral properties of compact operators. One can notice the similarity between this property and the fact that, if M and N are subspaces of a Banach space where M is closed and N is finite dimensional, then M + N is also closed.

Any compact operator is strictly singular, but not vice-versa.^[1]

Origins in integral equation theory

A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form
$(\lambda K + I)u=f \,$
(where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves much like as in finite dimensions. The spectral theory of compact operators then follows, and it is due to Frigyes Riesz (1918). It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limit point. Moreover, in either case the non-zero elements of the spectrum are eigenvalues of K with finite multiplicities (so that K − λI has a finite dimensional kernel for all complex λ ≠ 0).
An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Gårding inequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem into a Fredholm integral equation.^[2] Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.
The compact operators from a Banach space to itself form a two-sided ideal in the algebra of all bounded operators on the space. Indeed, the compact operators on a Hilbert space form a maximal ideal, so the quotient algebra, known as the Calkin algebra, is simple.

Compact operator on Hilbert spaces

Main article: Compact operator on Hilbert space
An equivalent definition of compact operators on a Hilbert space may be given as follows.
An operator $T$ on a Hilbert space $\mathcal{H}$

$T:\mathcal{H} \to \mathcal{H}$

is said to be compact if it can be written in the form

$T = \sum_{n=1}^\infty \lambda_n \langle f_n, \cdot \rangle g_n\,,$

where $f_1,f_2,\ldots$ and $g_1,g_2,\ldots$ are (not necessarily complete) orthonormal sets. Here, $\lambda_1,\lambda_2,\ldots$ is a sequence of positive numbers, called the singular values of the operator. The singular values can accumulate only at zero. If the sequence becomes stationary at zero, that is $\lambda_{N+k}=0$ for some $N\in\N, \quad k=1,2,\dots$ , then the operator has finite rank resp. a finite-dimenisional range and can be written as

$T = \sum_{n=1}^N \lambda_n \langle f_n, \cdot \rangle g_n\,.$

The bracket $\langle\cdot,\cdot\rangle$ is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.
An important subclass of compact operators are the trace-class or nuclear operators.

Completely continuous operators

Let X and Y be Banach spaces. A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence $(x_n)$ from X, the sequence $(Tx_n)$ is norm-convergent in Y (Conway 1985, §VI.3). Compact operators on a Banach space are always completely continuous. If X is a reflexive Banach space, then every completely continuous operator T : X → Y is compact.

Examples

For some fixed g ∈ C([0, 1]; R), define the linear operator T by

$(Tf)(x) = \int_0^x f(t)g(t) \, \mathrm{d} t.$

That the operator T is indeed compact follows from the Ascoli theorem.

More generally, if Ω is any domain in Rⁿ and the integral kernel k : Ω × Ω → R is a Hilbert—Schmidt kernel, then the operator T on L²(Ω; R) defined by

$(T f)(x) = \int_{\Omega} k(x, y) f(y) \, \mathrm{d} y$

is a compact operator.

By Riesz's lemma, the identity operator is a compact operator if and only if the space is finite dimensional.

See also

Notes

^ N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cambridge University Press.
^ William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000

References

Conway, John B. (1985). A course in functional analysis. Springer-Verlag. ISBN 3-540-96042-2

Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. (Section 7.5)

Kutateladze, S.S. (1996). Fundamentals of Functional Analysis. Texts in Mathematical Sciences 12 (Second ed.). New York: Springer-Verlag. p. 292. ISBN 978-0-7923-3898-7.

from：http://en.wikipedia.org/wiki/Compact_operator

举报/Report

我的更多文章

下载客户端阅读体验更佳