description/proof of that for vectors space with topology induced by metric induced by norm induced by inner product, if space is separable, it has no uncountable orthonormal subset
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of norm induced by inner product on real or complex vectors space.
- The reader knows a definition of metric induced by norm on real or complex vectors space.
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of separable topological space.
Target Context
- The reader will have a description and a proof of the proposition that for any vectors space with the topology induced by the metric induced by the norm induced by any inner product, if the space is separable, it has no uncountable orthonormal subset.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
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Statements:
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2: Note
Typically, although not necessarily,
3: Proof
Whole Strategy: Step 1: suppose that there was an
Step 1:
Let us suppose that there was an uncountable orthonormal subset,
Step 2:
For each
For each
Let
So,
So, we can take
So, the open balls are disjoint.
As
Then, any dense subset of
Then, the dense subset would need to have some uncountable points, because there were those uncountable open balls, a contradiction against
Step 3:
So, the supposition was wrong, and there is no uncountable orthonormal subset of