# Why Vector Spaces? Origins & Benefits

• genericusrnme
In summary, vector spaces and linear operators became the playground for quantum mechanics because they provide a mathematical framework that is well-suited for describing the state of a system in quantum mechanics. This idea was introduced by mathematicians such as Hilbert, Neumann, and Wigner, and it is based on the use of Hilbert spaces and the L2 norm. The use of vector spaces is also closely connected to the statistical interpretation of quantum mechanics and the idea that a wavefunction represents a complex root of a probability distribution.
genericusrnme
Just a quick question here;
Why is it that vector spaces and linear operators became the playground for quantum mechanics?
Most of the reasons I can think of seem to come out of things derived from the fact that we're using vector spaces..
Furthermore, who's idea was it to starting using vector spaces?

genericusrnme said:
Just a quick question here;
Why is it that vector spaces and linear operators became the playground for quantum mechanics?
Most of the reasons I can think of seem to come out of things derived from the fact that we're using vector spaces..
Furthermore, who's idea was it to starting using vector spaces?

The state of a system in quantum mechanics is a ket |ψ> which lives in a specific type of Vector space called Hilbert space. (Hilbert thought up Hilbert spaces [derr] but I think it might have been Neumann or Wigner who introduced them to physics--then again, I suspect Hilbert of jumping the gun on physics math but not admitting he did it to address physical problems. Remember he derived the Einstein Field Equations before Einstein.)

The reason we use Hilbert space is because Hilbert spaces come with the kind of norm that is natural for a wavefunction--the L2 norm. (This was invented by Lebesgue.) The L2 norm, in mathematics, is given by ∫ψ*(x)ψ(x)dx, which as you can see is a basis representation of the quantum physics expression <ψ|ψ>. The L2 norm can be loosely described as "euclidean," and this can be interpreted as reflecting the fact that QM operates on a background of euclidean space. (Not spacetime.)

Your question is definitely a deep one, and I encourage you to research the "Statistical Interpretation" of quantum mechanics. The question is entwined with the mysterious facts that a wavefunction is the complex root of a probability distribution, that measurable quantities are operators which act on kets, and that expectation values of measurements are given by <ψ|Q|ψ> = ∫ψ*(y)Qyψ(y)dy, (where Qy is the operator Q's expression in the y-basis. [I picked y to avoid x, which usually implies the position basis.])

## 1. What is a vector space?

A vector space is a mathematical concept that describes a set of objects, called vectors, that can be added together and multiplied by scalars (numbers). This set of vectors must satisfy certain properties, such as closure under addition and scalar multiplication, to be considered a vector space.

## 2. What are the origins of vector spaces?

The concept of vector spaces originated in the 19th century with the development of linear algebra. Mathematicians such as Georg Cantor, David Hilbert, and Hermann Grassmann laid the foundation for vector spaces through their work on vector and matrix operations. Vector spaces have since been applied in various fields, such as physics, engineering, and computer science.

## 3. Why are vector spaces important?

Vector spaces are important because they provide a powerful mathematical tool for representing and manipulating real-world data. They allow for the analysis and prediction of complex systems, from the motion of planets to the behavior of financial markets. Additionally, many mathematical concepts and techniques, such as eigenvalues and eigenvectors, are based on the properties of vector spaces.

## 4. What are the benefits of using vector spaces?

The use of vector spaces allows for the simplification and generalization of mathematical problems. By representing data in a vector space, complex systems can be broken down into simpler components, making them easier to analyze and understand. Vector spaces also provide a framework for solving problems that involve large amounts of data, making them a valuable tool in data science and machine learning.

## 5. How are vector spaces applied in different fields?

Vector spaces have a wide range of applications in various fields, including physics, engineering, computer science, and economics. In physics, vector spaces are used to represent physical quantities, such as force and velocity, and to study the behavior of complex systems. In engineering, vector spaces are used to model and analyze systems, such as circuits and mechanical structures. In computer science, vector spaces are used in data analysis and machine learning algorithms. In economics, vector spaces are used to model economic systems and analyze financial data.

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