vector spaces examples Archives - Time Travel, Quantum Entanglement and Quantum Computing https://stationarystates.com/tag/vector-spaces-examples/ Not only is the Universe stranger than we think, it is stranger than we can think...Hiesenberg Wed, 10 Feb 2021 19:27:58 +0000 en-US hourly 1 https://wordpress.org/?v=6.5.3 Vector Spaces Examples https://stationarystates.com/mathematical-physics/vector-spaces-examples/?utm_source=rss&utm_medium=rss&utm_campaign=vector-spaces-examples Sun, 24 Jan 2021 03:53:08 +0000 http://stationarystates.com/?p=69 The set of functions that take in a natural number n and return a REAL number.     Functions of ONE Real Variable (i.e. REAL  to REAL)     Of […]

The post Vector Spaces Examples appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

]]> The set of functions that take in a natural number n and return a REAL number.

    \[ \mathbb{R}^\mathbb{N} = \{f \mid f \colon \mathbb{N} \rightarrow \Re \} \]

Functions of ONE Real Variable (i.e. REAL  to REAL)

    \[ \mathbb{R}^\mathbb{R} = \{f \mid f \colon \Re \to \Re \} \]

Of what use are these vector spaces?

The fundamental theorem (of linear algebra) says that observables (in Quantum Mechanics) are Hermitian Matrices. Which means that the VECTORS form a complete set. That is, the vector space (corresponding to vectors of an observable) is a complete space.

Note that, this is the fundamental theorem of mathematics, not of Quantum Mechanics. Even though it applies perfectly to Quantum Mechanics. What is ever so mysterious about quantum mechanics is that the language of mathematics can be used throughout, without ever stepping into the physical meaning.

For instance, the Uncertainty relation is a basic mathematical property of two non-commuting matrices (observables). It has little to do with Quantum Physics, unless we make the connection to satisfy ourselves.

The post Vector Spaces Examples appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

]]>