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On Wikipedia, it says the dot product is valid for "any number of dimensions." Let's call it $n$.

$$u\cdot v = |u||v|\cos(\theta)$$

Is this still true if we let $n$ go to infinity?

EDIT: By coordinate vectors of infinte length, I mean coordinate vectors with infinite terms such as $u = <u_1,u_2,u_3, ...u_n, ...>$

transcendental
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1 Answers1

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First of all the dot product of two vectors $(v_1,\dots,v_n)$ and $(u_1,\dots,u_n)$ is defined by: $$ u \cdot v = \sum_i u_i v_i $$ and the angle $\theta$ is by consequence defined so that $$ \cos \theta = \frac{u\cdot v}{\lvert u\rvert \lvert v\rvert}. $$

As you can see the dot product can be extended to infinite dimensions if the series $$ \sum_i u_i v_i $$ converges. This is the case, for example, when both $u$ and $v$ satisfy: $$ \sum u_i^2 < \infty. $$ The sequences which satisfy this property are the elements of the space called $\ell^2$.

Emanuele Paolini
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  • Does series convergence imply that all terms can be calculated? Or why is it essential? – mavavilj Feb 25 '18 at 17:33
  • The series is required to converge so that the result of the dot product is a well defined finite real number. It might happen that you cannot compute the sum of a series, for example: 1 - 2 + 3 - 4 + 5 ... has not a well defined sum. – Emanuele Paolini Feb 27 '18 at 07:54