Prove that T is bounded and compact

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Reference no: EM132262948

Problem -

Define L²_w([0, 1]) = {f ∈ L²([0, 1]) : ₀∫¹|f(x)|²e^2/x dx < ∞}.

C^∞ members of L²_w should vanish to infinite order at x = 0.

(a) Show that L²_w is a separable Hilbert space with the natural inner product:

(f, g) = ₀∫¹f(x)(g(x))^-e^2/x dx.

Define T : L²_w → L²_w by

T(f) = 1/x₀∫^x f(x)dt.

(b) Prove that T is bounded and compact (Recall that T is also bounded on L² but it isn't compact on L²!)

Reference no: EM132262948

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