[Apronus Home] [Mathematics] 

Math ASCII Notation Demo
Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.
The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).
Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.
Show that a totally bounded metric space can be written as a finite union of open balls with radius e, for every e>0. 
______ 
Define an interesting metric on P(N). The one I picked up from the book "Topological Spaces" during vacation 2000. In this item, do not prove that it is indeed a metric. 
A,B c N if (A = B) then d(A,B) = 0 if (A != B) then d(A,B) = 1/min(A+B) A+B = (A\B)u(B\A) page 151 in golden gate 
A,B c N if (A = B) then d(A,B) = 0 if (A != B) then d(A,B) = 1/min(A+B); A+B = (A\B)u(B\A) Show that (P(N),d) is a metric space. 
First show that (*) d(A,B) < 1/m <=> A n [1,m] = B n [1,m]. Using this show the triangle inequality. page 151 in palace 
Let (X,d) and (Y,g) be metric spaces. Let f : X > Y. What does it mean that f is an isometry of X into Y? 
/\(a,b:X) [ d(a,b) = g(f(a),f(b)) ] (it follows that an isometry is 11) (I som' it rEE) 
Let (X,d) and (Y,g) be metric spaces. What does it mean that they are isometric? 
There exists a function f : X > Y such that (1) d(a,b) = g(f(a),f(b)) for all a,b from X (2) f(X) = Y (this f is bijective) isometric (I suh met' rik) 
Let (X,d) be a metric space. Consider the set B(X,R) of
bounded functions f:X>R. Let f = sup {f(x):x:X}. Show that B(X,R) is a metric space with d(f,g) = fg. 
_______ 
Let (X,d) be a metric space. Let B(X,R) be the set of bounded functions f:X>R. Equip it with the sup metric. What kind of metric space is this B(X,R) ? 
it is a complete metric space page 154 in the palace notebook 
Consider the set of all realvalued sequences a[n] such that
/\n 0<=a[n]<=1/n. Equip this set with the sup metric. What kind of metric space is this? 
It is compact. PROOF: Call it K and show that it is a closed
subset of the set of all bounded realvalued sequences.
Conclude that it is complete. Next show that K is totally
bounded. Here's how. Take any n:N. Let G = {g:K : /\k:N g(k):{0,1/n,2/n,...,(n1)/n,1}}. Show that G is finite. Show that the collection of balls with center g:G and ra 
Let X = (0,oo). Define d(x,y) = xy / (x*y). (X,d) is a
metric space. Is d similar to the Euclidean metric? 
Yes. Use d(x,y) = 1/x  1/y. 
Show that in a complete metric space (X,d), A c X: A is precompact <=> clo(A) is compact. Demonstrate that completeness is essential here. 
In every metric space A is precompact <=> clo(A) is precompact In a complete metric space, a closed set is complete. Precompact and complete implies compact. 