2025-05-06

1106: For Separable Hilbert Space and Subset, Closure of Subspace Generated by Subset Is Double-Orthogonal Complement of Subset

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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



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:
F: {R,C}, with the canonical field structure
(V,dist): ={ the separable Hilbert spaces }, with any inner product, ,, with the topology induced by dist
S: V
//

Statements:
(S)=S
//


2: Note


It is not "S=S": the subspace generated by S is required.

For example, when S={s} for a single s, S={s}, which does not equal S: while S is a vectors subspace, S is not even any vectors subspace at all.

Of course, when S is already a vectors subspace, (S)=S, and S=S holds.


3: Proof


Whole Strategy: Step 1: see that S=(S); Step 2: see that for each closed subspace of V, VV, V=V; Step 3: see that (S)=(S) and (S)=(S); Step 4: conclude the proposition.

Step 1:

Let us see that S=(S).

(S)=Span(S), by Note for the definition of sub-'vectors space' generated by subset of vectors space.

For each vS, v(S), because for each c1s1+...+cnsn(S), v,c1s1+...+cnsn=c1v,s1+...+cnv,sn=0+...+0=0.

For each v(S), vS, because S(S).

Step 2:

Let us see that for each closed subspace of V, VV, V=V, which is in fact a special case of this proposition, because (V)=V, because (V)=V as V is already a vectors subspace and V=V as V is already closed.

V is a separable Hilbert space, by the proposition that for any complete metric space, any closed subspace is complete and the proposition that for any separable topological space induced by any metric, any topological subspace is separable.

V has an orthonormal Schauder basis, B={b1,b2,...}, by the proposition that any separable Hilbert space has an orthonormal Schauder basis.

V has an orthonormal Schauder basis as an expansion of B, B={b1,b2,...}, by the proposition that for any separable Hilbert space and any orthonormal subset, the subset can be expanded to be an orthonormal Schauder basis.

V is nothing but the set of the elements that can be expressed as jvjbj: each element of V can be expressed such; each convergent jvjbj is a Cauchy sequence, so the convergence is in V, because V is complete.

V is nothing but the set of the elements that can be expressed as jvjbj where only B elements appear as bj s: likewise.

V is nothing but the set of the elements that can be expressed as jvjbj where only BB elements appear as bj s, because each element of V cannot have any B element and such each jvjbj is in V.

Then, V is nothing but the set of the elements that can be expressed as jvjbj where only B elements appear as bj s, because each element of V cannot have any BB element and such each jvjbj is in V.

So, V=V.

Step 3:

Let us see that (S)=(S).

(S)(S), because (S)(S) and each element of (S) is perpendicular to each element of (S), which is equal to or smaller than (S).

Let v(S) be any. For each v(S), v,v=0? For each ϵ-'open ball' around v, Bv,ϵV, there is a vBv,ϵ(S), by the proposition that the closure of any subset is the union of the subset and the accumulation points set of the subset. v,v=0. v,v=v,vv+v=v,vv+v,v=v,vv+0=v,vvvvv, by the Cauchy-Schwarz inequality for any real or complex inner-producted vectors space, <vϵ, but as ϵ is arbitrary, v,v=0.

So, v(S), which means that (S)(S).

So, (S)=(S).

So, (S)=(S).

Step 4:

By Step 2, (S)=(S): (S) is a vectors subspace, by 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.

By Step 1, (S)=S.

So, (S)=S.


References


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