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Let ?=ℝℝ be the real vector space of functions from ℝR to ℝR. Define ??={?∈? |...

Let ?=ℝℝ be the real vector space of functions from ℝR to ℝR. Define

??={?∈? | ?(−?)=?(?) ∀?∈ℝ}Ve={f∈V | f(−x)=f(x) ∀x∈R}

??={?∈? | ?(−?)=−?(?) ∀?∈ℝ}.Vo={f∈V | f(−x)=−f(x) ∀x∈R}.

a) Prove that ?=??⊕??V=Ve⊕Vo.

b) Give the decomposition of the function ?(?)=??f(x)=ex according to the above direct sum.

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