Metric Spaces - 1 Distance J MUSCAT 1 Metric Spaces Dr J...

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1 Distance J MUSCAT 1 Metric Spaces Dr J Muscat 2003 1 Distance Metric spaces can be thought of as very basic spaces having a geometry, with only a few axioms. They are generalizations of the real line, in which some of the theorems that hold for R remain valid. Some of the main results in real analysis are (i) Cauchy sequences converge, (ii) for continuous functions f (lim n →∞ x n ) = lim n →∞ f ( x n ), (iii) continuous real functions are bounded on intervals of type [ a, b ] and satisfy the intermediate value theorem, etc. At first sight, it is difficult to generalize these theorems, say to sequences and continuous functions of several variables f ( x, y ). That is the aim of this abstract course: to show how these theorems apply in a much more general setting than R . The fundamental ingredient that is needed is that of a distance or metric. This is not enough, however, to get the best results and we need to specify later that the distance be of a nicer type, called a complete metric. In what follows the metric space X will denote an abstract set, not nec- essarily R or R n , although these are of the most immediate interest. Definition A distance (or metric ) on a metric space X is a function d : X 2 R + ( x, y ) d ( x, y ) with the properties ( i ) d ( x, y ) = 0 x = y ( ii ) d ( y, x ) = d ( x, y ) ( iii ) d ( x, y ) d ( x, z ) + d ( y, z ) for all x, y, z X .
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1 Distance J MUSCAT 2 1.0.1 Example On N , Q , R , C and R N , one can take the standard Euclidean distance d ( x, y ) = | x - y | . Check that the three axioms for a distance are satisfied (make use of the fact that | a + b | | a | + | b | ). Note that for R n , the Euclidean distance d ( x , y ) = i = n i =1 | x i - y i | 2 . One can define distances on other more general spaces, e.g. the space of continuous functions f ( x ) of x [0 , 1] has a distance defined by d ( f, g ) = max x [0 , 1] | f ( x ) - g ( x ) | . The space of shapes (roughly speaking, subsets of R 2 having an area) have a metric d ( A, B ) = area of ( A B - A B ). In all these cases we get an idea of which elements are close together by looking at their distance. 1.0.2 Exercises 1. Show that if x 1 , . . . , x n are n points, then d ( x 1 , x n ) d ( x 1 , x 2 ) + . . . + d ( x n - 1 , x n ). 2. Verify that the metric defined on R 2 does indeed satisfy the metric axioms. 3. Show that if d is a metric, then so are the maps D 1 ( x, y ) = 2 d ( x, y ) and D 2 ( x, y ) = d ( x,y ) 1+ d ( x,y ) , but that d ( x, y ) 2 need not be a metric. Hence a metric space can have several metrics. 4. Show that if X , Y are metric spaces with distances d X and d Y , then X × Y is also a metric space with distance D ( x 1 y 1 , x 2 y 2 ) = d X ( x 1 , x 2 ) + d Y ( y 1 , y 2 ) . (Note that for R 2 , this metric is not the Euclidean one.) 5. Show further that ˜ D ( x 1 y 1 , x 2 y 2 ) = max( d X ( x 1 , x 2 ) , d Y ( y 1 , y 2 )) is also a metric for X × Y . 6. Prove that the function d ( m, n ) = | 1 /m - 1 /n | , on the natural numbers, is a metric.
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1.1 Balls J MUSCAT 3 7. Show that in fact the axioms (i) and (iii) imply that d ( x, y ) 0, and also axiom (ii).
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