Hilbert space - Wikipedia
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces.
Hilbert Space -- from Wolfram MathWorld
A Hilbert space is a vector space H with an inner product <f,g> such that the norm defined by |f|=sqrt(<f,f>) turns H into a complete metric space. If the metric defined by the norm is not complete, then H is instead known as an inner product space.
De nition 3.1(Hilbert space) A complete inner product space H is called a Hilbert space. C is complete, it's Hilbert space, and so is CN. l2 is a Hilbert space. L2(R) and L2([a; b]) are Hilbert spaces. (a) Consider a space E consisting of sequences fxng of complex numbers with only a number of nonzero terms.
There are really three `types' of Hilbert spaces (over C): The nite dimensional ones, essentially just Cn; with which you are pretty familiar and two in nite dimen-sional cases corresponding to being separable (having a countable dense subset) or not.
Hilbert space - Encyclopedia of Mathematics
2024年1月13日 · A Hilbert space is a natural extension of the ordinary three-dimensional space in Euclidean geometry, and many geometric concepts have their interpretation in a Hilbert space, so that one is entitled to speak about the geometry of Hilbert space.
There are really three `types' of Hilbert spaces (over C): The nite dimen-sional ones, essentially just Cn; for di erent integer values of n; with which you are pretty familiar, and two in nite dimensional types corresponding to being separa-ble (having a countable dense subset) or not.
A Hilbert space is an inner product space whose associated metric is complete. That is, a Hilbert space is an inner product space that is also a Banach space. For example, R n is a Hilbert space under the usual dot product:
A Hilbert space is a vector space V with an inner product satisfying (1.1) - (1.4) which is complete as a normed space (i.e., is a Banach space). Thus we have already shown L2(X; ) to be a Hilbert space for any positive measure . The inner product is. since then (1.3) gives kfk2.
We will introduce more mathematics later in the course as needed, but for now we will concentrate on the linear algebra of spaces of wave functions, which are called Hilbert spaces. These notes gather together and summarize most of what you will need to know of this subject, so we can proceed with other matters.
What Is Hilbert Space? - ScienceABC
2024年6月2日 · Hilbert Space is a mathematical space proposed by David Hilbert, a German mathematician. It is an extension of Euclidean space for infinite dimensions. Have you ever wondered how physicists are able to understand particles and waves?