description/proof of that for separable Hilbert space and subset, closure of subspace generated by subset is double-orthogonal complement of subset
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of Hilbert space.
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of separable topological space.
- The reader knows a definition of sub-'vectors space' generated by subset of vectors space.
- The reader knows a definition of closure of subset of topological space.
- The reader knows a definition of orthogonal complement of subset of vectors space with inner product.
- The reader admits the proposition that for any complete metric space, any closed subspace is complete.
- The reader admits the proposition that for any separable topological space induced by any metric, any topological subspace is separable.
- The reader admits the proposition that any separable Hilbert space has an orthonormal Schauder basis.
- The reader admits the Cauchy-Schwarz inequality for any real or complex inner-producted vectors space.
- The reader admits the proposition that for any vectors space with the topology induced by the metric induced by the norm induced by any inner product and any subspace, the closure of the subspace is a subspace.
- The reader admits the proposition that for any separable Hilbert space and any orthonormal subset, the subset can be expanded to be an orthonormal Schauder basis.
- The reader admits the proposition that the closure of any subset is the union of the subset and the accumulation points set of the subset.
Target Context
- The reader will have a description and a proof of the proposition that for any separable Hilbert space and any subset, the closure of the subspace generated by the subset is the double-orthogonal complement of the 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:
//
Statements:
//
2: Note
It is not "
For example, when
Of course, when
3: Proof
Whole Strategy: Step 1: see that
Step 1:
Let us see that
For each
For each
Step 2:
Let us see that for each closed subspace of
Then,
So,
Step 3:
Let us see that
Let
So,
So,
So,
Step 4:
By Step 2,
By Step 1,
So,