Download Kolmogorov, Fomin, Elements of the Theory of Functions and Functional Analysis. Volume 1 Metric and Normed Spaces PDF

TitleKolmogorov, Fomin, Elements of the Theory of Functions and Functional Analysis. Volume 1 Metric and Normed Spaces
File Size2.6 MB
Total Pages141
Table of Contents
	§1. The concept of set. Operations on sets
	§2. Finite and infinite sets. Denumerability
	§3. Equivalence of sets
	§4. Nondenumerability of the set of real numbers
	§5. The concept of cardinal number
	§6. Partition into classes
	§7. Mappings of sets. General concept of function
	§8. Definition and examples of metric spaces
	§9. Convergence of sequences. Limit points
	§10. Open and closed sets
	§11. Open and closed sets on the real line
	§12. Continuous mappings. Homeomorphism. Isometry
	§13. Complete metric spaces
	§14. Principle of contraction mappings and its applications
	§ 15. Applications of the principle of contraction mappings in analysis
	§16. Compact sets in metric spaces
	§17. ArzeIa's theorem and its applications
	§18. Compacta
	§19. Real functions in metric spaces
	§20. Continuous curves in metric spaces
	§21. Definition and examples of normed linear spaces
	§22. Convex sets in normed linear spaces
	§23. Linear functionals
	§24. The conjugate space
	§25. Extension of linear functionals
	§26. The second conjugate space
	§27. Weak convergence
	§28. Weak convergence of linear functionals
	§29. Linear operators
		Generalized Functions
	§30. Spectrum of an operator. Resolvents
	§31. Completely continuous operators
	§32. Ljnear operator equations. The Fredholm theorems
Document Text Contents
Page 70

§18] COMPACTA 59

of closed sets satisfying the condition

n(R '" 0 .. ) = o.
The condition that we can select a finite subcovering from {O .. } is equiva-

lent to the fact that the system of closed sets {R '" O .. } cannot have the
finite intersection property if it has a void intersection.

We shall now prove that Condition 1 is necessary and sufficient that R
be a compactum.

Necessity. Let R be a compactum and let {O .. } be an open covering of R.
We choose in R for each n = 1, 2, ... a finite (l/n)-net consisting of the
points ak (n) and we enclose each of these points with the sphere

S(ak(n), lin)

of radius lin. It is clear that for arbitrary n

R = UkS(A k (n), lin).

We shall assume that it is impossible to choose a finite system of sets
covering K from {O .. }. Then for each n we can find at least one sphere
S(ak(n)(n), lin) which cannot be covered by a finite number of the sets 0 ...
We choose such a sphere for each n and consider the sequence of their
centers {ak(n) (n)}. Since R is a compactum, there exists a point ~ E R which
is the limit of a subsequence of this sequence. Let Ofj be a set of {O .. } which
contains~. Since Ofj is open, we can find an E > 0 such that S{~, E} C Ofj .
We now choose an index n and a point ak(n) (n) so that p(~, ak(n) (n») < E/2,
lin < E/2. Then, obviously, S(ak(n)(n), lin) c Ofj, i.e. thesphereS(ak(n)(n),
lin) is covered by a single set Ofj . The contradiction thus obtained proves
our assertion.

Sufficiency. We assume that the space R is such that from each of its
open coverings it is possible to select a finite subcovering. We shall prove
that R is a compactum. To do this it is sufficient to prove that R is complete
and totally bounded. Let E > O. Take a neighborhood O(x, E) about each
of the points x E R; we then obtain an open covering of R. We choose from
this covering a finite subcovering O(Xl , E), ... ,O(xn , E). It is clear that the
centers Xl, ••• , Xn of these neighborhoods form a finite E-net in R. Since
E > 0 is arbitrary, it follows that R is totally bounded. Now let {Sn} be a
sequence of nested closed spheres whose radii tend to zero. If their inter-
section is void, then the sets R '" S" form an open covering of R from which
it is impossible to select a finite subcovering. Thus, from Condition 1 it
follows that R is complete and totally bounded, i.e. that R is compact.

THEOREM 5. The continuous imO{Je of a compactum is a compactum.
Proof. Let Y be a compactum and let Y = cp(X) be its continuous image.

Further, let {O .. } be an open covering of the space Y. Set U .. = cp-l(O .. ).

Page 71


Since the inverse image of an open set under a continuous mapping is open,
it follows that {U a} is an open covering of the space X. Since X is a com-
pactum, we can select a finite subcovering U1 , U2, ... , Un from the
covering {Ua }. Then the corresponding sets 0 1 , O2 , ••• , 0" form a finite
subcovering of the covering {O a}.

THBOREM 6. A one-to-one mapping of a compactum which is continuous in
one direction is a homeomorphism.

Proof. Let cp be a one-to-one continuous mapping of the compactum X
onto the compactum Y. Since, according to the preceding theorem, the
continuous image of a compactum is a compactum, the set cp(M) is a com-
pactum for an arbitrary closed M C X and consequently cp(M) is closed in
Y. It follows that the inverse image under the mapping cp -1 of an arbitrary
closed set M C X is closed; i.e. the mapping cp -1 is continuous.

RI<;MARK. The following result follows from Theorem 6: if the equation

(3) dy/dx = f(x, y),

where the function f(x, y) is continuous in a closed bounded region G
which contains the point (xo, Yo) for every Yo belonging to some closed
interval [a, b], has a unique solution satisfying the initial condition y(xo) =
Yo , then this solution is a continuous function of the initial value Yo .

In fact, since the function f(x, y) is continuous in a closed bounded
region, it is bounded and consequently the set P of solutions of equation
(3) corresponding to initial values belonging to the closed interval [a, b]
is uniformly bounded and equicontinuous. Moreover, the set P is closed.
In fact, if {cp,,(x)} is a sequence of solutions of equation (3) which con-
verges uniformly to a function cp(x), then cp(x) is also a solution of
equation (3) since, if

cp,,' = f(x, CPn(x)),

then passing to the limit as n ~ 00, we obtain

cp' = f(x, cp(x)).

We have

cp(xo) = lim cp,,(xo) E [a, b].

In virtue of Arzela's theorem and Theorem 1 of this section it follows
from this that P is a compactum.

We set the point cp(xo) of the closed interval [a, b] into correspondence
with each solution cp(x) of equation (3). By assumption this correspondence
is one-to-one. Moreover, since

Page 140

Product of operator::! 98
Proper subset 1

Quadratic metric 19

Radius of a sphere 23
Range of variation of a function 13
Real functions in metric spaces 62 ff.
Rectifiable curves 69
Reflexive space 89
Reflexivity 12
Regular point 110
Residual spectrum 111
Resolvent 110

Schwartz 105
Schwarz inequality 17, 19
- conjugate space 88
- countability axiom 28
Segment 74
Semi continuous function 64
Set 1
-, Cantor 32
-, closed 26
-, compact 51
-, compact in itself 58
-, convex 74
-, dense 25
-, denumerable 4, 7
-, derived 26
-, everywhere dense 25
-, finite 4
-, infinite 4, 8
-, open 27
-, nondenumerable 5
-, totally bounded 51
-, void 1
Simplex 76
-, complete 37
-, conjugate 82
-, connected 29
-, Hausdorff 31
-, linear 71
-, metric 16
--, n-dimensional R" (Euclidean) 17
-, n-dimensional Ro n 17
-, n-dimensional Rp" 20
-, normed 71

D1DEX 129

- of continuous function::; C 18
- of continuous functions C[a, b) 17,72
- of continuous functions C2[a, b) 19, 72
-- of continuous functions with quad-

ratic metric 19
-- of real numbers Rl 16, 72
- of sequences c 72
- of sequences l2 18, 72
- of sequences lp 22
- of sequences 1/1. 72
-, reflexive 89
-, separable 25
-, topological 30
- with countable basis 28
-, continuous 111
- of an operator 110
-, point 111
-, residual 111
Subset 1
-, proper 1
Subspace 73
- of operators 97
- of sets 1,2
Symmetric difference of "eta :2
Symmetry 12
System of sets with finite intersection

property 58

Theorem on nested spheres :39
Topological space 30
- boundedness 51
- variation 64
Totally bounded set 51
Transitivity 12

Uniform bounded ness of n family of
functions 54

Union 1
Upper limit 64

Variation, see Total variation
Vertices of a simplex 76
-- integral equation 50
--- operator 114

Weak convergence 90,93
Weak* convergence 93

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