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Let’s say we have some $n$-tuple $x=(x_1,\dots,x_n)$. Could we take its $\mathscr{l}^2$ norm? I mean what’s the difference between a vector and a tuple anyway? Well, this question led me to believe that the $\mathscr{l}^2$ norm is not defined for tuples. However, my friend insists that it does exist and that is similarly defined as $$ \|x\|=\left(\sum_i x_i\right)^{1/2}. $$ But, aren’t tuples and vectors basically the same thing? Of course, one represents displacement relative to zero and the other a fixed point, but in this context, are they the same?

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A realization came over me - coordinate vectors are not the same thing as vectors: a vector is an element of a vector space where as a coordinate vector is a linear combination of basis vectors.