Check whether the identity map on an infinite dimensional space is compact.
Functional analysis is all about studying infinite-dimensional spaces and the maps between them, so this question is right up its alley. In the realm of functional analysis, the identity map on an infinite-dimensional space is a classic case to consider.
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Let T be a linear map defined on X by
Show that T is a bounded linear map such that T = .1
Define Eigen Spectrum of a bounded linear operator on a Banach space. Show that the eigen spectrum of the operator T on l2 given by ) (T ,......... ) =
is empty .
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 A be an operator on a Hilbert space .H Show if Ax A x ∗ = for every ,H x ∈ then A is normal. Is it converse true? Justify
Show that Q defined on by
dt)t(xt is a bounded linear functional. Calculate
.
The space l3 is a Hilbert space
Define 3 3 A : C → C by A z( , z , z ) iz( , e z , z ) 2 3