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Question Number 197667 by SANOGO last updated on 25/Sep/23

Answered by Frix last updated on 26/Sep/23

$$X,Y\in {\mathbb{R}}^3,r\in \mathbb{R}$$$$$$$$\left(1\right)N\left(X\right)=0\Rightarrow X=0✓$$$$X\ne 0\iff x_1\ne 0\vee x_2\ne 0\vee x_3\ne 0\iff N\left(X\right)\ne 0$$$$$$$$\left(2\right)N\left(r\times X\right)=\mid r\mid \times N\left(X\right)✓$$$$N\left(r\times X\right)=\sqrt{r^2{x}_1^2+r^2{x}_2^2}+\mid {rx}_3\mid =\mid r\mid \sqrt{x_1^2+x_2^2}+\mid r\mid \times \mid x_3\mid =\mid r\mid \times N\left(X\right)$$$$$$$$\left(3\right)N(X+Y)⩽N\left(X\right)+N\left(Y\right)✓$$$$\sqrt{{(x_1+y_1)}^2+{(x_2+y_2)}^2}+\mid x_3+y_3\mid ⩽\sqrt{x_1^2+x_2^2}+\mid x_3\mid +\sqrt{y_1^2+y_2^2}+\mid y_3\mid$$$$\left(a\right)$$$$\sqrt{{(x_1+y_1)}^2+{(x_2+y_2)}^2}⩽\sqrt{x_1^2+x_2^2}+\sqrt{y_1^2+y_2^2}$$$${(x_1+y_1)}^2+{(x_2+y_2)}^2⩽x_1^2+x_2^2+y_1^2+y_2^2+2\sqrt{(x_1^2+x_2^2)(y_1^2+y_2^2)}$$$$x_1{y}_1+x_2{y}_2⩽\sqrt{x_1^2{y}_1^2+x_1^2{y}_2^2+x_2^2{y}_1^2+x_2^2{y}_2^2}$$$$x_1^2{y}_1^2+2x_1{x}_2{y}_1{y}_2+x_2^2{y}_2^2⩽x_1^2{y}_1^2+x_1^2{y}_2^2+x_2^2{y}_1^2+x_2^2{y}_2^2$$$$0⩽x_1^2{y}_2^2-2x_1{y}_2{x}_2{y}_1+x_2^2{y}_1^2$$$$0⩽{(x_1{y}_2-x_2{y}_1)}^2✓$$$$\left(b\right)$$$$\mid x_3+y_3\mid ⩽\mid x_3\mid +\mid y_3\mid$$$${(x_3+y_3)}^2⩽x_3^2+y_3^2+2\mid x_3\mid \mid y_3\mid$$$$2x_3{y}_3⩽2\mid x_3{y}_3\mid ✓$$$$$$$$\Rightarrow N\left(X\right)\mathrm{est}\mathrm{une}\mathrm{norme}$$$$$$$$X\in {\mathbb{R}}^n,n\in \mathbb{N}\Rightarrow$$$$N_1\left(X\right)\mathrm{est}\mathrm{une}\mathrm{norme}\wedge N_2\left(X\right)\mathrm{est}\mathrm{une}\mathrm{norme}$$$$\Rightarrow N_1\left(X\right)\mathrm{est}\acute{\mathrm{e}quivalente}\stackrel{`}{\mathrm{a}}{N}_2\left(X\right)$$

Commented by SANOGO last updated on 26/Sep/23

$$thankyousir$$

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