function ?: ℝ → ℝ is called even if ?(−?) = ?(?) for all ? ∈ ℝ, and a function ?: ℝ → ℝ is called odd if ?(−?) = −?(?). Let ?? denote the set of real-valued even functions on ℝ, and let ?? denote the set of real-valued odd functions on ℝ. a) Show that ?? is a subspace of ℳ(ℝ, ℝ). b) Show that ?0 is a subspace of ℳ(ℝ, ℝ). c) Show that ℳ(ℝ, ℝ) = ?? ⊕ ??. Please use clear mathematical

function ?: ℝ → ℝ is called even if ?(−?) = ?(?) for all ? ∈ ℝ, and a function ?: ℝ → ℝ is called odd if ?(−?) = −?(?). Let ?? denote the set of real-valued even functions on ℝ, and let ?? denote the set of real-valued odd functions on ℝ. a) Show that ?? is a subspace of ℳ(ℝ, ℝ). b) Show that ?0 is a subspace of ℳ(ℝ, ℝ). c) Show that ℳ(ℝ, ℝ) = ?? ⊕ ??. Please use clear mathematical

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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