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"In Hilbert space two functions that have the same square integral are considered equivalent. Technically, vectors in Hilbert space represent equivalence classes of functions."

But that means that if we take for example

f(x) = 1 | 0 < x < 1

and

g(x) = [itex]\sqrt{1/10}[/itex] | 0 < x < 10

both f(x) and g(x) have the same square integral ∫|f(x)|^2 dx = ∫|g(x)|^2 dx = 1

But how can they be considered "equivalent" - if they are wavefunctions for a particle, for example, then they represent completely different probability predictions! f(x) says you can find a particle with equal probability in [0,1], while for g(x) it's [0,10]