- #1
TZenith
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1. X : Banach space
Z : closed subspace of X
Prove or disprove that X* ⊆ Z*
where Z* and X* are dual space of Z and X, respectively.
2. X : normed space and f : X → R : linear functional.
Assume that ∃a∈X and r∈(0,1] such that f(B(a,r))=R(Real numbers)
where B(a,r) is open ball. Prove that f is discontinuous
Proof. 2) I think it must suppose f is continuous and find a contradiction with f(B(a,r))=R
But I can't find some contradiction.
1) We think it doesn't satisfy X* ⊆ Z*. But we will show X* ⊆ Z* and find some contradiction. Let f ∈ X*. Thus f : X → R is bounded linear functional. We must to show that f : Z → R is bounded linear functional or find some contradiction.
Z : closed subspace of X
Prove or disprove that X* ⊆ Z*
where Z* and X* are dual space of Z and X, respectively.
2. X : normed space and f : X → R : linear functional.
Assume that ∃a∈X and r∈(0,1] such that f(B(a,r))=R(Real numbers)
where B(a,r) is open ball. Prove that f is discontinuous
Proof. 2) I think it must suppose f is continuous and find a contradiction with f(B(a,r))=R
But I can't find some contradiction.
1) We think it doesn't satisfy X* ⊆ Z*. But we will show X* ⊆ Z* and find some contradiction. Let f ∈ X*. Thus f : X → R is bounded linear functional. We must to show that f : Z → R is bounded linear functional or find some contradiction.