1. Let S ⊆ H be any non-empty subset of S. S⊥={x∈H:∀s∈S⟨x,s⟩=0}
Let x,y∈S⊥. Then for any s∈S we have ⟨x+y,s⟩=⟨x,s⟩+⟨y,s⟩=0+0=0 and for any scalar λ we have ⟨λx,s⟩=λ⟨x,s⟩=0. Therefore, S⊥ is a linear.
If {xn}n=1+∞⊂S⊥ converges to x∈H, then for any s∈S we have ⟨x,s⟩=limn→+∞⟨xn,s⟩=0.
This implies that S⊥ is a closed subspace of H.
2. Let L ⊆ H be a linear manifold (i.e. closed linear subspace) of H. For all x∈L and y∈L⊥ we have ⟨x,y⟩=0, therefore, by the definition, L⊂L⊥⊥.
If x∈/L then let xL be an orthogonal projection of to L. It can be obtained as follows.
L is a closed linear subspace of H, therefore L is a Hilbert space. Let {v1,v2,...} - the orthonormal basis of L, cn=⟨x,vn⟩, xn=∑k=0nckvk. Then if k<n then ⟨xn,vk⟩=⟨x,vk⟩ and ⟨x−xn,vk⟩=0. From the Pythagorean theorem ∣x∣2=∣(x−xn)+xn∣2=∣x−xn∣2+∣xn∣2, from where ∣xn∣2≤∣x∣2 . From the other side, ∣xn∣2=∑k=0n∣ck∣2 , therefore the partial sums of the series ∑k=0+∞∣ck∣2 are bounded and the sequence xn is fundamental in L, hence, it converges to some vector xL∈L. Since ⟨x−xn,vk⟩=0 , then ⟨x−xL,vk⟩=limn→+∞⟨x−xn,vk⟩=0 for any k. Then x−xL∈L⊥ and ⟨x−xL,x⟩=⟨x−xL,x⟩−⟨x−xL,xL⟩=⟨x−xL,x−xL⟩=∣x−xL∣2>0
This shows that x∈/L⊥⊥ and therefore, L=L⊥⊥.
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