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David Hilbert
David Hilbert (German: David Hilbert [ˈdaːvɪt ˈhɪlbɐt] (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert was born in 1862 in Königsberg (present-day Kaliningrad, Russia) and died in Göttingen, Germany, in 1943. He is revered as a great mathematician and scientist for inventing a large number of ideas (e.g., invariant theory, axiosterized geometry, Hilbert space). [1]
David Hilbert
Hilbert Space
Hilbert space refers to the complete inner product space, which is synonymous. Non-complete inner product spaces are also known as pre-Hilbert spaces.
Then obviously there is the following relationship:
That is, Hilbert space is a special kind of inner product space, and its particularity is reflected in its completeness, because an inner product space is not necessarily a complete space.
There are two concepts involved, then, namely: "complete space" and "inner product space". The intersection of the two is the "complete inner product space". Explained separately below.
Complete space
In mathematical analysis , a complete space is also known as a complete metric space or Cauchy space. If all Cauchy sequences in a metric space converge to a point in that space, the space is called a complete space. [2]
Two concepts are involved in this definition, namely "Metric space" and "Cauchy sequence".
Measurement space
In mathematics , a metric space is a set with a distance function that defines the distance between all elements within the set. This distance function is called a measure on the set. The metric space that best fits people's intuitive understanding of reality is the three-dimensional Euclidean space. [3]
"Distance" here is an abstract concept that refers not only to the straight-line distance between two points, but also to vector distances, function distances, surface distances, and so on. Defined as:
Let it be a non-empty set, pair any two points, and under the action of the metric, there is a real number corresponding to these two points and satisfies:
Positive: , and if and only if true;
Symmetry: ;
Trigonometric inequality: +.
Then it is called a distance (measure) in it, which is called a measurement space for a measure.
Cauchy sequence
In mathematics , a Cauchy sequence , Cauchy , Cauchy sequence , or fundamental column is a sequence of numbers whose elements get closer and closer as the ordinal increases. Any convergent sequence must be a Cauchy column, and any Cauchy column must be a bounded sequence. [4]
Completeness
As mentioned earlier, "If all Cauchy sequences in a metric space converge to a point in that space, the space is called a complete space. ”
Real and rational numbers can be used as concrete examples.
Sequences defined by real numbers are complete in the usually defined sense of distance.
Sequences defined by rational numbers are not complete in the usually defined sense of distance. For example, a sequence of rational numbers:
++, i.e. . The Babylonian method [5] can be used to prove that its results converge to .
Having said all this, to use a popular but not rigorous phrase to express it: in the space usually seen, the real number space is the complete space.
Inner product space
Refers to a vector space (or "linear space", synonymous with the addition of an "algorithm" (or "structure"), which is also known as "Scalar product" or "Dot product". The inner product concatenates a pair of vectors with a pure quantity, allowing us to talk strictly about the "angles" and "lengths" of the vectors, and further talk about the orthogonality of the vectors. [6]
This in turn involves the concept of "Vector space".
Moreover , the inner product space has a norm naturally defined based on the inner product of the space itself , and it satisfies the parallelogram theorem , that is , the inner product can induce a norm , so the inner product space must be " norm space " . This in turn involves the concept of "Normed vector space".
Step by step, let's talk about vector spaces (or "linear spaces", which are synonymous).
vector space
A general vector space is defined as follows: a vector space laid out in a field (e.g., a field of real numbers, a field of complex numbers) is a collection of vectors and assigned to the addition between the set vector and the vector: +; and the multiplication between scalar and vector: . The sum of the vectors is +, and the product of the vector and scalar is . Vector addition and scalar multiplication in vector space satisfy:
Addition commutative law: ++ ;
Law of Additive Union: (+) ++ (+) ;
Vector units: there is a unique such that + ;
Inverse element: There is a unique existence that makes + ;
Vector allocation law: for , ++;
Scalar allocation law: for , ++;
Conjugating law: for ;
Scalar units: For .
Thus , vector space is essentially an additive commutative group appended to an operation that specifies the product of each scalar and vector as a vector , and that vector . It can be seen that the definition of vector space does not contain multiplication between vectors and vectors.
This is also the reason why the inner product is an additional condition that distinguishes the inner product space from the general vector space. This is why the inner product space contains three operations: addition between vectors and vectors, multiplication between scalars and vectors, and multiplication between vectors and vectors.
On the basis of understanding the vector space, we will turn around and add the concept of the norm space and the relationship between these spaces. Since the normatization space is defined on the basis of the vector space, it is also called the linear normization space, referred to as the normating space. Note that, as mentioned earlier, a vector space is a linear space, and the two are synonymous.
(Linear) normed space
Norms are often used to measure the length or size of each vector in a vector space (or matrix). It is defined as:
Let is a vector space that is laid across a field (e.g., a field of real numbers, a field of complex numbers), a function acts on, and the condition is met:
Positive: Yes and if and only if;
Flush: Yes, yes;
Trigonometric inequality: Yes, with ++.
The symmetry is a norm on which the vector space that defines the norm is called a (linear) norm-setting space.
By comparing the normization space with the metric space above, we can see that the difference between "norm" and "distance" is:
Distances ( or " measures " ) are defined on arbitrary non-empty sets , while norms are defined on vector spaces;
In vector spaces, norms can induce distances (or "metrics"), and vice versa does not hold, which also means that the norm space must belong to the metric space;
The "flushing" of the norm suggests that the norm can be seen as an enhanced concept of distance.
The following illustration shows the containment relationship between several spaces:[7]
summary
Having said all this, it feels a bit messy, so let's sum up the relationship between the various spaces:
Vector space + norm operation = (linear) norm space
(Linear) normed space + inner product operation = inner product space
(Linear) normed space + completeness = Banach space
Inner product space + completeness + finite dimension = Euclidean space
Inner product space + completeness = Hilbert space
A final addition: Hilbert space is a generalization of finite-dimensional Euclidean space, making it not limited to the case of real numbers and finite dimensions, but without losing completeness (unlike non-Euclidean spaces in general).
And from the above relationship, it can be seen that Hilbert space can be seen as a Banach space with an increase in the inner product operation.
Reference
David_Hilbert
Complete metric space
Metric space
Cauchy sequence
Babylonian method
Inner product space
Normed vector space