A Very Cool Analysis Fact
by djmathman, Nov 8, 2016, 1:43 AM
Man, Math Studies Analysis is introducing us to some very very neat things. Here's just one example.
First, a few preliminary definitions.
DEFINITION 1: Let
be a set. We say that a function 
is a metric if for any 
the following three properties hold:
is a metric space. Sometimes we write 
as the metric if we wish to emphasize the metric space we're working in.
There are many many different types of metric spaces. Below are some examples:
DEFINITION 2: Let and 
be two metric spaces. We say that an isometric embedding of into is a function 
such that ![\[d_X(x,y) = d_Y(\varphi(x),\varphi(y))\text{ for all }x,y\in X.\]](//latex.artofproblemsolving.com/1/b/4/1b414dbfec42f996ebe688dfaa20ee33dc970b51.png)
If 
is actually bijective, we say that and are isomorphic and write 
.
There are also weaker notions of this kind of correspondence (homeomorphisms, uniform homeomorphisms, bi-Lipschitz homeomorphisms) but that's a topic for another day.
Now we come to another question: what do we mean by "topological"? Well, there are some special properties that some metric spaces have that are worth studying. Analogously, think about sets within this metric space; we have a simple notion of different types of sets within a metric space - e.g. open/closed/compact - and sometimes these properties play nicely with respect to transformations (e.g. the image of a continuous function on a compact set is compact). We can play the same game here. There are many definitions, but only one is of use to us right now.
DEFINITION 3: Let be a metric space. We say that is separable if has a countable dense subset 
. Here, dense has a precise meaning, but for simplicity let's just say that given some point 
, we can always find elements of close enough to even if we shrink our radius of interest.
This notion of separability is a bit wonky at first, so here are a few examples of where this property shows up.
Now here's the kicker. Remember how I said above that sometimes metric spaces can look very different but share some very surprising commonalities? Well, we can take this one step further.
THEOREM (Banach-Mazur): Let be a metric space. Then is separable if and only if is isomorphic to some subset of ![$C^0([0,1];\mathbb{R})$](//latex.artofproblemsolving.com/3/e/b/3ebc123f414074fabb9212c193174a213f8fe262.png)
.
This means that we can create any rediculously abstract metric space that we want, but as long as it satisfies separability, we can find some copy of it sitting inside the set of continuous functions from![$[0,1]$](//latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
to 
.
Pretty cool, huh?
First, a few preliminary definitions.
DEFINITION 1: Let
be a set. We say that a function 
is a metric if for any 
the following three properties hold:
- POSITIVITY:
, and 
iff 
. - SYMMETRY:
. - TRIANGLE INEQUALITY:
.
,
iff 
.
.
.
is a metric space. Sometimes we write 
as the metric if we wish to emphasize the metric space we're working in.
is a semimetric. It's worth noting that one can always convert a semimetric into a metric by creating an equivalence relation 
such that 
iff There are many many different types of metric spaces. Below are some examples:
- Trivially,
, 
, and all form metric spaces under the usual absolute value metric, namely 
. - For any set , define the discrete metric via
![\[d(x,y) = \begin{cases}1&x\neq y,\\0&x=y.\end{cases}\]](//latex.artofproblemsolving.com/4/b/d/4bd08a92bc1312d4e89ddb25260277fc7b5808fb.png)
Then is a metric space. - For any metric spaces and , set
to be the collection of bounded functions 
. Here we say that 
is bounded if there exists some constant 
such that 
for all 
. This is a metric space when equipped with the metric ![\[d_{\mathcal{B}(X;Y)}(f,g) = \sup_{x\in X}d_Y(f(x),g(x)).\]](//latex.artofproblemsolving.com/6/2/9/629b634d5503dd56686506409dc10f6f154148d3.png)
We can also look at 
and 
, which are the collection of continuous functions and (continuous and bounded) functions respectively from to . The same metric applies. - The Hilbert Cube
is defined by ![\[H^\infty = \{\{x_n\}_{n=0}^\infty: x_i\in[0,1]\text{ for all }i\geq 0\}.\]](//latex.artofproblemsolving.com/4/f/9/4f9599cb362f80d8f9d90f7e1413fdaba0c9538a.png)
This is a metric space when endowed with the metric ![\[d_{H^\infty}(x,y) = \sum_{k=0}^\infty\dfrac{|x_k-y_k|}{2^k}.\]](//latex.artofproblemsolving.com/8/f/c/8fc623b0d9fef7fadf5b166a02d94b8880d879ce.png)
,
,
.![\[d(x,y) = \begin{cases}1&x\neq y,\\0&x=y.\end{cases}\]](http://latex.artofproblemsolving.com/4/b/d/4bd08a92bc1312d4e89ddb25260277fc7b5808fb.png)
Then 
to be the collection of bounded functions 
.
is bounded if there exists some constant 
such that 
for all 
.![\[d_{\mathcal{B}(X;Y)}(f,g) = \sup_{x\in X}d_Y(f(x),g(x)).\]](http://latex.artofproblemsolving.com/6/2/9/629b634d5503dd56686506409dc10f6f154148d3.png)
We can also look at 
and 
,
is defined by ![\[H^\infty = \{\{x_n\}_{n=0}^\infty: x_i\in[0,1]\text{ for all }i\geq 0\}.\]](http://latex.artofproblemsolving.com/4/f/9/4f9599cb362f80d8f9d90f7e1413fdaba0c9538a.png)
This is a metric space when endowed with the metric ![\[d_{H^\infty}(x,y) = \sum_{k=0}^\infty\dfrac{|x_k-y_k|}{2^k}.\]](http://latex.artofproblemsolving.com/8/f/c/8fc623b0d9fef7fadf5b166a02d94b8880d879ce.png)
DEFINITION 2: Let
and 
be two metric spaces. We say that an isometric embedding of into is a function 
such that ![\[d_X(x,y) = d_Y(\varphi(x),\varphi(y))\text{ for all }x,y\in X.\]](http://latex.artofproblemsolving.com/1/b/4/1b414dbfec42f996ebe688dfaa20ee33dc970b51.png)
If 
is actually bijective, we say that and are isomorphic and write 
.There are also weaker notions of this kind of correspondence (homeomorphisms, uniform homeomorphisms, bi-Lipschitz homeomorphisms) but that's a topic for another day.
Now we come to another question: what do we mean by "topological"? Well, there are some special properties that some metric spaces have that are worth studying. Analogously, think about sets within this metric space; we have a simple notion of different types of sets within a metric space - e.g. open/closed/compact - and sometimes these properties play nicely with respect to transformations (e.g. the image of a continuous function on a compact set is compact). We can play the same game here. There are many definitions, but only one is of use to us right now.
DEFINITION 3: Let
be a metric space. We say that is separable if has a countable dense subset 
. Here, dense has a precise meaning, but for simplicity let's just say that given some point 
, we can always find elements of close enough to even if we shrink our radius of interest.This notion of separability is a bit wonky at first, so here are a few examples of where this property shows up.
- If is finite or countable, then is clearly separable: just let your dense countable subset be the entire metric space!
- Note that is dense in since we can always approximate any real number with a sequence of converging rationals. Since is also countable, we can thus deduce that is separable.
- If is separable and
, then is separable. (This is a nontrivial result; it appeared as one of our homework problems a few weeks back.) - The Hilbert cube from above is separable.
- Conversely, if is an infinite set and is a metric space with
, then is not separable.
,
,Now here's the kicker. Remember how I said above that sometimes metric spaces can look very different but share some very surprising commonalities? Well, we can take this one step further.
THEOREM (Banach-Mazur): Let
be a metric space. Then is separable if and only if is isomorphic to some subset of ![$C^0([0,1];\mathbb{R})$](http://latex.artofproblemsolving.com/3/e/b/3ebc123f414074fabb9212c193174a213f8fe262.png)
.This means that we can create any rediculously abstract metric space that we want, but as long as it satisfies separability, we can find some copy of it sitting inside the set of continuous functions from
![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
to 
.Pretty cool, huh?
This post has been edited 1 time. Last edited by djmathman, Nov 8, 2016, 1:51 AM
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