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Andre' Quanta
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Why we require the separability of Hilbert spaces in Quantum Mechanics?
Andre' Quanta said:Why we require the separability of Hilbert spaces in Quantum Mechanics?
The separability of Hilbert Spaces refers to the property of a Hilbert Space being able to be spanned by a countable set of vectors. In other words, it means that the space contains a countable dense subset.
The completeness of a Hilbert Space is a necessary condition for separability. A Hilbert Space must be complete in order to have a countable dense subset, which is a defining feature of separability. However, not all complete Hilbert Spaces are separable.
Yes, one example of a separable Hilbert Space is the space of square-integrable functions on the interval [0,1] with the inner product defined as the integral of the product of two functions. This space is separable because it can be spanned by the set of polynomials with rational coefficients, which is a countable set.
The concept of separability is important in functional analysis because it allows for the study and analysis of infinite-dimensional spaces using tools and techniques from finite-dimensional linear algebra. It also has applications in various fields such as physics, engineering, and statistics.
No, not every subspace of a separable Hilbert Space is necessarily separable. A subspace can only be separable if it contains a countable dense subset. If the subspace is not dense, then it cannot be separable. However, if the subspace is also complete, then it must be separable.