Series y Funciones

40

Metric

Spaces

1.6

space if we choose

|

" enlarged" to the real l ine R w hich is complete. A n d this "complet i o n " R o f Q is such that Q is dense (cf. 1.3-5) m R . I t isquite

7. Let X be the set of all positive integers and dim. n) = |m"' - n~'\.

i m p o r t a n t t hat an a r b i t r a r y incomplete metric space c an be " c o m -

Show that [X, d) is notcomplete.

p l e t e d " i n a similar fashion, as we shall see. For a convenient precise
f o r m u l a t i o n w e use the f o l l o w i n g t wo related concepts, w hich also have

8. fSpace C[a,b]) Show that the subspace V c C [ a , b] consisting of all

v arious other applications.

x £ C[a, b] such that xtal = x(i>) is complete.

0
/

w

1 .6-1

e

referred to the followingtheorem of calculus. If a sequence
l x_) of continuous functions on [a, b] converges on [a, b] and the
convergence is uniform on [a. b], then the limit function x is continuous on [a. b]. Prove thistheorem.

' J - 10. (Discrete metric)

Show that a discrete metric space (d.

1.1-8)

D efinition

( Isometric

« i for all / = I , 2. • • • , where x„ = (£"'>

a

n

d

isometrics paces).

( a) A m apping T of X i nto X is said to be isometric
isometr)' if T p reserves d istances, that is. if for all ,t, y 6 X ,

is

d(Tx,

L et

o r an

T y) = d(x. y).

where Tx a nd T y are the images of x a nd y . r espectively.

1 1 . (Space s) Show that in the space s (cf. 1.2-1) we have x, —— X if and
ft"

mapping,

X = ( X , d) a nd X = ( X . d) be metricspaces. T h e n :

complete.

only if

41

Spaces

W e k now that the rational l ine Q is not complete (cf. 1.5-7) b ut can be

dix, y) = ¡are tan x - a r c tan y|.

n

of Metric

1. 6 Completion o f Metric Spaces

6. Show that the set of all real numbers constitutes an incomplete metric

^ Cjjy '

Completion

(b) T h e space X is said to be isometric w i t h t he...