Question Number 120524 by SOMEDAVONG last updated on 01/Nov/20
![1/.Given E=C([0,1],R) are mapping set [0,1]→R. (a). Show that E is vector space. (b).For f∈E ,let ∣∣f∣∣_1 =∫_0 ^1 ∣f(x)∣dx . Show that (E,∣∣.∣∣_1 ) are normed space. (Helpe me please)](Q120524.png)
$$\mathrm{1}/.\mathrm{Given}\:\mathrm{E}=\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\mathbb{R}\right)\:\mathrm{are}\:\mathrm{mapping}\:\mathrm{set}\:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}. \\ $$$$\left(\mathrm{a}\right).\:\mathrm{Show}\:\mathrm{that}\:\mathrm{E}\:\mathrm{is}\:\mathrm{vector}\:\mathrm{space}. \\ $$$$\left(\mathrm{b}\right).\mathrm{For}\:\:\mathrm{f}\in\mathrm{E}\:,\mathrm{let}\:\mid\mid\mathrm{f}\mid\mid_{\mathrm{1}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mid\mathrm{f}\left(\mathrm{x}\right)\mid\mathrm{dx}\:. \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{E},\mid\mid.\mid\mid_{\mathrm{1}} \right)\:\mathrm{are}\:\mathrm{normed}\:\mathrm{space}. \\ $$$$\:\:\left(\mathrm{Helpe}\:\mathrm{me}\:\mathrm{please}\right) \\ $$