Norm under Real Numbers

Topics: Algebra - Vector Space - Inner Product

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In ${\mathbb{R}}^n$ (as the usual vector space), we define the Norm of a vector $u=(u_1,u_2,\dots ,u_n)$ as:

$$\Vert u\Vert =\sqrt{u_1^2+u_2^2+\cdots +u_n^2}$$

This norm satisfies the previous properties. Also note that $\Vert u{\Vert }^2=⟨u,u⟩$.

Thanks to the law of cosines, we also have that:

$$⟨u,v⟩=\Vert u\Vert \cdot \Vert v\Vert \cos \theta$$