Two properties of Hilbert Spaces that always confused me (due to the mathematical rigor needed to understand them) is
(a) Completeness, and
(b) Separable Space
(apart from the easy to understand concept of LVS and Inner product)
I have created a few slides to help understand these two properties by comparing with physical examples.
Completeness : A Mathematical space is COMPLETE if there is a (Cauchy) sequence of elements/vectors that get closer and closer together, and finally converges into another element that belongs to the original space.
Separable Space: A Hilbert Space is a LVS, whose elements/vectors are SEPARABLE, which means the elements are building blocks of that space that can each be counted as separate from the other, even though the total number of elements themselves add up to infinite.
Hope this helps in the understanding of these two properties of Hilbert spaces, that apply equally to the Quantum states belonging to it's own Hilbert space.
(a) Completeness, and
(b) Separable Space
(apart from the easy to understand concept of LVS and Inner product)
I have created a few slides to help understand these two properties by comparing with physical examples.
Completeness : A Mathematical space is COMPLETE if there is a (Cauchy) sequence of elements/vectors that get closer and closer together, and finally converges into another element that belongs to the original space.
Separable Space: A Hilbert Space is a LVS, whose elements/vectors are SEPARABLE, which means the elements are building blocks of that space that can each be counted as separate from the other, even though the total number of elements themselves add up to infinite.
Hope this helps in the understanding of these two properties of Hilbert spaces, that apply equally to the Quantum states belonging to it's own Hilbert space.
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