Consider the vector space V = C([0, 1]) of complex-valued functions that are continuous on [0, 1]. Show that for every alpha α ∈ C and f ∈ C([0,1]), we have (a) ⟨f − αg, f − αg⟩ = ⟨f, f⟩ − 2Re(conjugate of α⟨f, g⟩) + |α|2⟨g, g⟩ ≥ 0.For g ≠ 0, set α = ⟨f, g⟩/⟨g, g⟩, which is defined since ⟨g, g⟩ = ||g||22 ≠ 0. With this value of α, prove the Cauchy-Schwartz inequality for L2:|⟨f, g⟩| ≤ ||f||L2 ||g||L2.

To establish the Cauchy- Schwarz inequality for the functions in C[0,1]= the space of complex valued…

Answered: Consider the vector space V = C([0, 1])…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Consider the vector space V = C([0, 1]) of complex-valued functions that are continuous on [0, 1]. Show that for every alpha α ∈ C and f ∈ C([0,1]), we have
(a) ⟨f − αg, f − αg⟩ = ⟨f, f⟩ − 2Re(conjugate of α⟨f, g⟩) + |α|²⟨g, g⟩ ≥ 0.
For g ≠ 0, set α = ⟨f, g⟩/⟨g, g⟩, which is defined since ⟨g, g⟩ = ||g||²₂ ≠ 0. With this value of α, prove the Cauchy-Schwartz inequality for L²:

|⟨f, g⟩| ≤ ||f||_L² ||g||_L².

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