Transcription of Metric Spaces - UiO
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Metric Spaces - UiO
Chapter 1 Metric SpacesMany of the arguments you have seen in several variable calculus are almostidentical to the corresponding arguments in one variable calculus, especiallyarguments concerning convergence and continuity. The reason is that thenotions of convergence and continuity can be formulated in terms of distance ,and that the notion of distance between numbers that you need in the onevariable theory, is very similar to the notion of distance between points orvectors that you need in the theory of functions of severable variables. Inmore advanced mathematics, we need to find the distance between morecomplicated objects than numbers and vectors, between sequences, setsand functions. These new notions of distance leads to new notions of con-vergence and continuity, and these again lead to new arguments suprisinglysimilar to those we have already seen in one and several variable a while it becomes quite boring to perform almost the same argu-ments over and over again in new settings, and one begins to wonder if thereis general theory that cover all these examples is it possible to developa general theory of distance where we can prove the results we need onceand for all?
distance we get when we stop at a third pont z along the way, i.e. d(x,y) ≤ d(x,z)+d(z,x) It turns out that these conditions are the only ones we need, and we sum them up in a formal definition. Definition 1.1.1 A metric space (X,d) consists of a non-empty set X and a function d : …
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