Show that r is compact, Show that R is compact and Show

Show that r is compact

Assignment Help Mathematics

Reference no: EM133334532

Problem: Recall that the range of a linear operator A : Η → Η is defined by

RanA := { y ∈ Η:y = Ax, for some x ∈ Η}.

A bounded linear operator R:Η→Η > is a finite rank operator if its range is finite dimensional.

a) Assume that the dimension of RanR is N and pick any basis {n_j}^N_j=1 in RanR.

Show that one can write

R(f) = Σ_j=1^Nα_j(f)n_j

where α_i, j = 1,......, N are bounded linear functionals.

b) Show that R is compact.

Reference no: EM133334532

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