Characterise all bounded linear functionals on a Hilbert space.
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Let H be a Hilbert space. For any subset A of ,H define . A ⊥ If ,H A ⊆ B ⊆ then show that:
i) B1⊆ A
ii) A ⊆ A ⊥⊥
State conditions on A⊥⊥ =A
Let Y A : X0 ⊆ X → be a closed operator where X and Y are Banach spaces. Define x x Ax , x X . A = + ∈ 0 Then show that the norm A ⋅ is complete.
Check whether a finite dimensional normed linear space is reflexive? Justify your answer.
Let be a norm on a linear space .X if X ,x y∈ and
then show that
for all .0 s ≥ t,0 ≥ .
State the principle of uniform boundedness. Use it to show that a set E in a normed space X is bounded if )E(f is bounded in K for every f ∈ X′.
Prove that l1⊂ l2 If: T :(l2 ||.||)(l1,||.||2)
(l1,||.||2) is a compact operator, show that: T:,||.||
||.||2 is also compact.
Show how a real linear functional u on a complex linear normed space gives rise to a complex linear functional .f What is the relation between the boundedness of u and that of f ?