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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.