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Hilbert spaces

Introduction to Hilbert vector spaces

If you are a fan of quantum mechanics, you have surely heard of Hilbert spaces. This type of vector space is crucial in all branches of quantum physics since the wave function and the operators that act on it are defined on a Hilbert space.

Definition of vector spaces and generalization to Hilbert spaces

The Hilbert spaces are a type of abstract vector spaces that generalize the concept of Euclidean space. To really understand what they are we have to understand what a vector space is.

A vector space is a type of algebraic structure defined on a set of elements provided with two operations: an internal operation that is the addition of vectors and an external operation that is the multiplication of vectors by scalars.

If about this vector space we define a new internal operation , the famous scalar product or inner product, then we are providing this vector space with a geometry. This happens since the dot product allows us to define angles and distances between the elements of the vector space through the norm.

When the elements of this vector space with the defined geometry are the set of real numbers then we are talking about a Euclidean vector space, since it complies with Euclid's geometric axioms. Some examples are the line (one-dimensional) or the plane (two-dimensional).

Hilbert spaces are born to generalize Euclidean spaces to infinite dimensions. Furthermore, the elements of this type of vector spaces can be functions and therefore we have a functional space.

This type of vector spaces are more abstract than Euclidean spaces since they allow us to define a geometry (distances and angles) between functions, for example, complex ones. Intuitively, it is more difficult to understand a distance or angle between two functions, which is why we say that they are abstract vector spaces.

Examples of Hilbert vector spaces

Just with the definition of Hilbert spaces it is difficult to really understand what it is about. The abstract form of mathematics makes many concepts easier if examples are used. Next we will see two common examples of Hilbert spaces.

Euclidean Spaces

The first basic example of a Hilbert space is Euclidean vector spaces. As we have explained, Hilbert spaces are a generalization of Euclidean spaces, therefore, every finite Euclidean vector space is also a Hilbert space (not vice versa).

An example is the three-dimensional space defined on the set of real ones.

Functional spaces

The set of functions defined between two vector spaces can themselves form a Hilbert vector space structure defined by the inner product or scalar product between two functions of the set. These spaces are of vital importance for fields such as quantum mechanics, artificial intelligence, partial differential equations or Fourier analysis.