Hilbert Space
Hilbert space is a fundamental mathematical concept that extends the idea of traditional geometric spaces into an infinite-dimensional context. It serves as a vector space defined by an inner product that is both complete and has no gaps, allowing for sophisticated calculations that go beyond simple algebra or calculus. The concept originated from Euclidean spaces, which deal primarily with two- and three-dimensional configurations, and was significantly developed by mathematician David Hilbert in the early 20th century.
Hilbert spaces are crucial in fields such as quantum mechanics, where they provide the mathematical framework for understanding the behavior of particles at microscopic scales. This includes applications in solving complex equations like Schrödinger's equation, which describes how quantum systems evolve over time. Additionally, while all Hilbert spaces are classified as Banach spaces, not all Banach spaces meet the criteria to be considered Hilbert spaces. The interplay of Hilbert spaces with analysis, algebra, and topology enhances the ability to analyze and calculate properties of objects in mathematics and physics, making it an invaluable tool for researchers in these areas.
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Hilbert Space
Hilbert space is a mathematical concept in linear geometry that defines an infinite-dimensional space. In other words, it takes geometric concepts that are limited to dealing with two- and three-dimensional spaces and expands them so that they can be used with an infinite number of dimensions. Instead of using straight algebra or calculus, the scientist has an expanded range of calculations and can analyze concepts in multiple dimensions. A Hilbert space is a vector space in which the inner product is defined and complete. The concept is related to both Euclidean spaces and Banach spaces. It is an important concept in quantum mechanics and mathematical analysis.
Background
The concept of Hilbert space was derived from Euclidean spaces. A Euclidean space is defined as a vector space with a positive definite inner product. A vector is the mathematical representation of something—generally a mathematic quantity or a phenomenon—that has both direction and magnitude, or size. In mathematical diagrams, vectors generally appear as arrows that convey this information. The direction is indicated by the way the arrow is pointing, and the size is indicated by the length of the arrow. A vector space refers to a space where the vectors can be added and multiplied by scalars, or points in the space that are represented by real numbers and that have either direction or magnitude but not both.
When these can be multiplied and/or added to result in a positive definite inner product, the vector space is known as a Euclidean space. Euclidean spaces are named after the Greek mathematician Euclid, who flourished about 300 BCE. Euclid is known as the "father of geometry." He first defined the concept of a Euclidean space.
In the first decade of the twentieth century, a German mathematician named David Hilbert, who is considered one of the preeminent mathematicians of the century, enhanced a number of concepts related to geometry. For example, he expanded the concept of Euclidean spaces from those limited to two- and three-dimensional spaces to those with infinite dimensions. About twenty years later, Hungarian-born American mathematician John von Neumann became the first to add Hilbert's name to the concept, calling them Hilbert spaces. Two other mathematicians, Erhard Schmidt, a Russian, and Frigyes Riesz, a Hungarian, also experimented with the concepts and theories of Hilbert space.
Overview
Hilbert space is defined by mathematicians as a complete inner product space. An inner product space is one in which the total of multiplying and adding the points contained in the vector yields a defined product with specific properties. A "complete" inner product space has no gaps between the real numbers that are part of its sequences. These sequences must not turn into irrational numbers, or the inner product is no longer considered complete.
A Hilbert space has an infinite number of dimensions; this is what differentiates it from other types of spaces, such as Euclidean spaces or Banach spaces. Banach spaces are named after Polish mathematician Stefan Banach. The concept was included in a doctoral dissertation that Banach wrote and submitted in 1920. The idea was attributed to Banach even though others introduced similar concepts about the same time. These mathematical spaces are vector spaces with norms, or mathematical functions that specifically limit the vectors to a particular positive size or length. All Hilbert spaces are Banach spaces, but not all Banach spaces fit the criteria of Hilbert spaces.
Each of these types of spaces is a theoretical depiction of an abstract subject using a sequence of numeric representations for the size of a series of objects (such as atoms, electrons, stars, etc.) and their distance from one another. Using the equations relative to the concept (multiplying and adding the scalars), it is possible to calculate the movement and speed of the objects.
The ability to perform linear geometric functions without limitations on the dimensions makes the concept of Hilbert space a valuable tool for analysis. The concept allows several key branches of mathematics—analysis, algebra, and topology—to work together. Topology pertains to the properties and spatial relationships of geometric figures that are not affected when the sizes or shapes of the figures are changed. The ability to calculate sequences to infinity facilitates analysis in ways linear geometry or algebra alone cannot do.
The concepts of Hilbert space became very important to physicists trying to solve mathematical formulas—especially those related to quantum mechanics. Quantum mechanics is the branch of science that deals with topics including the movement and speed of movement of tiny particles such as electrons and atoms. The ability to calculate, define, and analyze distance and size through Hilbert space equations is important because the equations used to calculate movement and speed of larger, macroscopic objects are not accurate when applied to microscopic matter. They are often used by physicists working on Schrödinger's equation, which is named after Austrian physicist Erwin Schrödinger and is used to determine the energy levels and related wave functions of quantum mechanical subjects such as electrons and atoms.
Although von Neumann named Hilbert space and popularized it in a book he wrote in 1932, he later came to question its usefulness. In the book, Mathematical Foundations of Quantum Mechanics, von Neumann noted that Hilbert space is based on Euclid space and that the way the vectors were used in Hilbert space was redundant. However, the book did not fully describe why von Neumann changed his view on Hilbert space.
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