Hilbert Space - infinite dimensions

[Mikhail Liebestein]

David Hilbert 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics(particularly proof theory).

 

Increasingly physicists are turning to the idea of what is called Hilbert Space to explain the nature and reality of the universe(s).

 

In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. 

 

------------------

Every inner product 〈., .〉 on a real or complex vector space H gives rise to a norm ∣∣.∣∣as follows:

 

∥x∥=√〈x,x〉

 

 

We call H a Hilbert space if it is complete with respect to this norm. Completeness in this context means that every Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero. Every Hilbert space is thus also a Banach space (but not vice versa).

All finite-dimensional inner product spaces (such as Euclidean space with the ordinary dot product) are Hilbert spaces. However, the infinite-dimensional examples are much more important in applications.

 

--------

 

The infinite dimension aspect gets interesting as many observations in quantum physics can only be explained if the universe branches every time a quantum decoherence event arises, it's a bit like 2 universes branching each times a Schrödinger's cat experiment is observed.

 

The Universe essentially branches, with the different reality, in the same overall universe, just occupying a different part of the infinite dimension Hilbert Space.