Thanks, but I find it a bit boring. So many different aspects only to describe a linear function. I hope that at least the list at the end is of some help to look up definitions in a short time. I hope the next part will be a bit more exciting, i.e. more examples than theory. However, one needs the vocabulary first.jedishrfu said:
A Hilbert space is a mathematical concept that is used to describe an infinite-dimensional vector space. It is named after the German mathematician David Hilbert and is used in functional analysis, linear algebra, and quantum mechanics.
A Hilbert space is an infinite-dimensional space, while a Euclidean space is a finite-dimensional space. This means that a Hilbert space can have an infinite number of dimensions, while a Euclidean space has a limited number of dimensions. Additionally, a Hilbert space is equipped with a special inner product, which allows for the concept of orthogonality and the existence of a basis of orthonormal vectors.
Hilbert spaces are used to describe the state of a quantum system in quantum mechanics. The vectors in a Hilbert space represent the possible states of the system, and the inner product between vectors represents the probability of transitioning from one state to another. This allows for the mathematical description of quantum phenomena such as superposition and entanglement.
Unlike finite-dimensional spaces, Hilbert spaces cannot be visualized in the traditional sense. This is because they have an infinite number of dimensions, making it impossible to accurately represent them in a visual form. However, certain aspects of Hilbert spaces, such as basis vectors and orthogonality, can be visualized using mathematical representations.
Hilbert spaces have connections to various other mathematical concepts, such as Banach spaces, which are complete normed spaces. They are also closely related to functional analysis, which deals with the study of functions on infinite-dimensional spaces. Additionally, Hilbert spaces are used in areas such as signal processing, control theory, and quantum computing.
