Someone proposes f,g = _0^1 f'(x)^* g'(x)

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

Someone proposes ⟨f,g⟩=∫01f′(x)∗g′(x) dx\langle f,g\rangle = \int_0^1 f'(x)^* g'(x)\,\mathrm{d}x as an inner product on the space P3\mathcal{P}_3 of real polynomials of degree at most 3 on [0,1][0,1], and argues: "conjugate symmetry and linearity in the second argument are obvious, and positivity holds because ∫01(f′)2 dx≥0\int_0^1 (f')^2\,\mathrm{d}x \ge 0 for every real ff, so this is a genuine inner product." ((a)) Check each of the three inner-product axioms (conjugate symmetry, linearity in the second argument, and positive-definiteness ⟨f,f⟩>0\langle f,f\rangle> 0 for f≠0f \ne 0) for this pairing on P3\mathcal{P}_3. ((b)) State whether the pairing is a genuine inner product on P3\mathcal{P}_3, and if not, name explicitly a nonzero vector that shows which axiom fails.

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