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I understand Unit concept as a measurement concept, so when we say Unit vector we mean it’s the smallest vector(and it’s length always equal one) and we can measure other vectors in terms of it.

For Normalization, the concept of normalization as I understand it is “return” the values of that thing to be in $[0,1]$ interval, so when we normalize a vector we divide it by it’s length to get a vector that inside that interval, and it’s length not necessarily equals one.

I’m I right?

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If you normalize a vector in a normed vector space $(V,||.||)$ you actually apply the mapping: $$V\backslash\{0\}\rightarrow S^1,v\mapsto\frac{1}{||v||}v $$ where $S^1=\{v\in V:||v||=1\}$ is the unit sphere. Clearly $$||\frac{1}{||v||}v||=\frac{||v||}{||v||}=1$$ for all $v\in V\backslash\{0\}$.

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  • $\begingroup$ So, in conclusion: in normed vector space there is no difference between “unit” and “normalization”? $\endgroup$ Sep 14, 2020 at 16:05
  • $\begingroup$ Or let' s say normalization is the mapping that maps every non-zero vector on its correspondent unit vector $\endgroup$ Sep 14, 2020 at 17:45

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