F-which-is-the-set-of-funtions-from-R-to-R-is-a-vectorial-space-and-G-a-part-of-F-is-the-set-of-odd-functions-such-as-G-f-F-x-R-f-x-f-x-1-Show-that-G-is-sub-vector-space-of-F-in-R-

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Question Number 89530 by mathocean1 last updated on 17/Apr/20

$$\mathrm{F}\mathrm{which}\mathrm{is}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{funtions}\mathrm{from}\mathbb{R}\mathrm{to}\mathbb{R}$$$$\mathrm{is}\mathrm{a}\mathrm{vectorial}\mathrm{space}\mathrm{and}\mathrm{G}\left(\mathrm{a}\mathrm{part}\mathrm{of}\mathrm{F}\right)\mathrm{is}$$$$\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{odd}\mathrm{functions}\mathrm{such}\mathrm{as}$$$$\mathrm{G}=\{f\in \mathrm{F}/\mathrm{\forall }\mathrm{x}\in \mathbb{R},f\left(x\right)=-f(-x)\}$$$$1)S\mathrm{how}\mathrm{that}\mathrm{G}\mathrm{is}\mathrm{sub}\mathrm{vector}\mathrm{space}\mathrm{of}\mathrm{F}$$$$\mathrm{in}\mathbb{R}.$$

Commented by arcana last updated on 18/Apr/20

$$\mathrm{The}\mathrm{function}\mathrm{zero}f\left(x\right)=0,\mathrm{\forall }x\in \mathbb{R},\mathrm{so}$$$$f\left(x\right)=0=f(-x)\Rightarrow 0\in \mathrm{G}.\mathrm{G}\ne \varphi .$$$$\mathrm{G}\subseteq \mathrm{F}.$$$$\mathrm{let}\mathrm{be}f,g\in \mathrm{G}$$$$(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)$$$$=-f(-x)+[-g(-x)]$$$$=-(f+g)(-x),\mathrm{\forall }x\in \mathbb{R}$$$$f+g\in \mathrm{G}.$$$$\mathrm{for}\alpha \in \mathbb{R},\left(\alpha f\right)\left(x\right)=\alpha (-f(-x))=-\left(\alpha f\right)(-x)$$$$\Rightarrow \alpha f\in \mathrm{G}.$$$$$$

Commented by mathocean1 last updated on 18/Apr/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}!$$

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