LikeMath said:I am sure that my questions are stupid. If we have a Hilbert space H, what do we mean by the closed subspace of H.
Also, Does every Hilbert space have an identity? :P.
What do you mean with identity?? It has a 0, which is the additive identity...
LikeMath said:I mean 1.
micromass said:That doesn't really help me. What is 1 supposed to mean??
LikeMath said:1 is the multiplicative identity
LikeMath said:Oops, thank you, that is why the question is stupid.
A Hilbert space is a mathematical concept in functional analysis that refers to an infinite-dimensional vector space with a defined inner product. It is named after the German mathematician David Hilbert and is widely used in fields such as physics, engineering, and mathematics.
In mathematics, an identity element is an element that, when combined with another element using a specific operation, leaves the other element unchanged. In the context of a Hilbert space, having an identity means that there exists an element that, when used in an inner product with any other element, does not change the other element.
Yes, every Hilbert space has an identity element. This is because the concept of an identity element is a fundamental property of Hilbert spaces and is defined as part of the structure of a Hilbert space.
The identity element of a Hilbert space is typically represented by the number 1, as it is the multiplicative identity element in most Hilbert spaces. In some cases, it may also be represented by the zero vector, as it is the additive identity element in many Hilbert spaces.
No, there are no exceptions to this rule. The concept of an identity element is a fundamental property of Hilbert spaces, and it is a necessary component for the proper functioning of a Hilbert space. Therefore, all Hilbert spaces must have an identity element.