# Hilbert Space Notes

Hilbert Space Notes. The study of functions and the qualities they possess falls within the purview of the mathematical discipline known as functional analysis. Hilbert space is an essential notion in this domain. David Hilbert, a German mathematician, is credited with being the one who initially presented the idea in the early 20th century. As a result, the notion bears his name.

## Main Theme of Hilbert Space

A Hilbert space is what’s known as a complete inner product space, which essentially implies that it’s a vector space that’s also a full metric space and comes packed with an inner product. This inner product meets certain criteria, including symmetry, linearity in the first argument, and positive definiteness, amongst others. Because the space is complete, any Cauchy sequence of vectors that is contained inside the space has a limit that is likewise contained within the space.

The concept that Hilbert spaces provide a framework that can be used for the investigation of functions that may have an unlimited number of dimensions is the primary tenet that underpins Hilbert spaces. In particular, they are helpful for the analysis of functions that are not always continuous but that are “well-behaved” in some sense.

One of the most prominent examples of a Hilbert space is the space of square-integrable functions, which consists of all functions whose square can be integrated across some domain. This space was named after David Hilbert. In the fields of quantum physics and signal processing, this space is often referred to as L2 and is used.

Another significant example is the space of sequences that may be added together to form a square, which is represented by the symbol l2 in mathematics.

## Hilbert spaces are used in many other areas

Furthermore, Hilbert spaces are used in a wide variety of different subfields within mathematics. They are an effective method for researching functions that have an unlimited number of dimensions. In addition, they have a multitude of essential applications in the scientific and technological fields of today.