Let act on complex-valued functions on

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Inner products, orthogonality and the adjoint

Problem

Let T=d/dxT = \mathrm{d}/\mathrm{d}x act on complex-valued functions on [0,2][0,2], with inner product ⟨f,g⟩=∫02f∗(x)g(x) dx\langle f,g\rangle = \int_0^2 f^*(x)g(x)\,\mathrm{d}x. ((a)) Integrate by parts to write ⟨f,Tg⟩\langle f, Tg\rangle as a boundary term minus ⟨f′,g⟩\langle f', g\rangle. ((b)) Evaluate the boundary term for f(x)=xf(x) = x and g(x)=x2g(x) = x^2. ((c)) Check the result directly by computing ⟨f,Tg⟩\langle f,Tg\rangle and ⟨f′,g⟩\langle f',g\rangle separately.

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