What is a Hilbertum and an Euclidean room?

2021-08-29 22:10:21 ROSEMARY

I have already looked at the internet, but could only understand little things. Can someone explain to me, so that it would understand a 10 stalemate?


Hello Lila1112,

A Hilbertraum is a complete scalar product space.

To explain what is , I have to make a difference :

Body (English Field )

Below is a set k, defined in addition and multiplication completed under both operations in which both Operations associative and commutative are, for both a neutral element (0 or 1) and there is a negative and for each item a reciprot and for each out of 0 - and the distributive law applies. The amount ℚ of the rational numbers and the amount ℝ of the real numbers are body.

Vector space over a body K

This is a set V, for which an addition and multiplication with Einem Scalar (an element of K) is defined. Under both operations V must be completed, and for the multiplication of a vector sum with a scalar, the distributive law applies.

K is of course also vector space (dimension 1) via itself.

An example of a three-dimensional vector space is ℝ³ whose elements than

(1) x> = (X; y; z) or (x₁; x₂; x₃)

, wherein the X¡ are real numbers.

Scalar product space

Belongs one understands one Vector room with in which a scalar product is additionally defined between two elements. As the label says, the scalar product itself is a scalar, e.g. = x₁ ∙ y₁ + x₂ ∙ y₂ + x₃ ∙ y₃. The scalar product also induces a standard , e.g.

(2) | x> | = √ {} = x₁² + x₂² + x₃²,

a non-negative number, which describes the "length" of the vector in our example.

Incidentally, there are many different possibilities To define standards that do not have to be due to a scalar product. The standard represented by (2) is called the Euclidean Standard , and a vector space where this standard is defined means Euclidean .

Cokey consequences and completeness

A sequence is an illustration of the amount ℕ of the natural numbers on a body or a vector space, ie, each natural number is assigned, for example, a real number or a vector. A cauchy consequence is a sequence whose members are roughly spoken with each other.

More precisely, for eachPositive real number ε, no matter how small ε, there is a location n₀, so that for two natural numbers m ≥ n₀ and n ≥ n₀ | vₙ - vₘ | < ε ist. Dabei sind vₙ und vₘ Elemente des entsprechenden Körpers oder Vektorraums.

A result is convergent to a limit value V when there is a N₀ for each ε, so that for each natural number n ≥ N₀ | Vₙ - V | < ε ist.

completely means a vector space V when each cauchy sequence of elements from V are converged within

V. It also applies to body. ℝ, for example, is completely, ℚ, on the other hand, because of rational numbers can also converge against an irrational number, eg, 3, 3.1, 3,14, 3,141, 3,1415, 3,14159, ... converged against π ,


Whole Platt: In the Euclidean space, for example, to three-dimensionally arranged variables as vectors are displayed spatially. The HIbertro extends this to infinity, so that multi-dimensional variables (eg from large matrices) can be shown and can be viewed spatially (eg to be able to determine angles between individual vectors, from which, for example, statements about observed relationships between variables can be derived, s. Correlation).

What is a Hilbertum and an Euclidean room?