In linear algebra and its application professor Gilbert Strang writes:
"A much better idea is to keep the familiar definition of length, using a sum of squares, and to include only those vectors that have a finite length:
Length squared = $||v||^2 = v_1^2 +v_2^2 + v_3^2 + ...$
The infinite series must converge to a finite sum. this leaves (1, 1/2, 1/3,...) but not (1,1,1,...). The vectors with finite length can be added $(\|v + w\| \le \|v\| + \|w\|)$ and multiplied by scalars, so they form a vector space. It is the celebrated hilbert space."
this leaves (1, 1/2, 1/3,...) but not (1,1,1,...)? does this make sense. to me partial sums of this (1,1,1,...) do not converge
What you've described is an example of a Hilbert space. Our vectors need not be lists of coordinates, and our measurement of length doesn't need to be the usual sum of squares (that is, we can define different inner products). However: you can indeed say that $(1,1,\dots)$ fails to be an element of the Hilbert space that you've described.