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cianfa72

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- About the definition of ##L^2## square integrable functions and the Hilbert vector space dimensionality

Hi,

I'm aware of the ##L^2## space of square integrable functions is an Hilbert space.

I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral, one can show that the set of square integrable functions on any ##(X,\Sigma)## measurable space actually fulfills the axioms of vector space.

Then one introduce the inner product ##< ,>## that induces a norm and a metric/distance. Next step proves that every Cauchy sequence converges to an element of the vector space itself (i.e. the completeness condition is met).

Therefore ##L^2##-space turns out to be an infinite dimensional Hilbert space.

Is its dimension either countable or uncountable infinity ?

I'm aware of the ##L^2## space of square integrable functions is an Hilbert space.

I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral, one can show that the set of square integrable functions on any ##(X,\Sigma)## measurable space actually fulfills the axioms of vector space.

Then one introduce the inner product ##< ,>## that induces a norm and a metric/distance. Next step proves that every Cauchy sequence converges to an element of the vector space itself (i.e. the completeness condition is met).

Therefore ##L^2##-space turns out to be an infinite dimensional Hilbert space.

Is its dimension either countable or uncountable infinity ?

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