1. X : Banach space(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Prove or disprove about Dual space

**Physics Forums | Science Articles, Homework Help, Discussion**