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543: １３２人目の素数さん [] 2023/03/18(土) 23:18:41.86 ID:M09HE8oG

>>535
>https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space
>Reproducing kernel Hilbert space

追加引用 （一部google訳）
https://upload.wikimedia.org/wikipedia/commons/6/65/Different_Views_on_RKHS.png
図は、RKHS を表示するための関連するさまざまなアプローチを示しています。

RKHS ではない関数のヒルベルト空間を構成することは、完全に単純ではありません。[1]ただし、いくつかの例が見つかっています。[2] [3]

It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS.[1] Some examples, however, have been found.[2][3]

L2 spaces are not Hilbert spaces of functions (and hence not RKHSs), but rather Hilbert spaces of equivalence classes of functions (for example, the functions
f and g defined by f(x)=0 and
g(x)=1_Q are equivalent in L2). However, there are RKHSs in which the norm is an L2-norm, such as the space of band-limited functions (see the example below).

An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every x in the set on which the functions are defined, "evaluation at x" can be performed by taking an inner product with a function determined by the kernel. Such a reproducing kernel exists if and only if every evaluation functional is continuous.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1677671318/543

643: １３２人目の素数さん [] 2023/03/21(火) 17:43:41.86 ID:8s9PZXQ2

>>642
つづき

https://en.wikipedia.org/wiki/Commutative_algebra
Commutative algebra
This article is about a branch of algebra. For algebras that are commutative, see Commutative algebra (structure).
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z ; and p-adic integers.
Commutative algebra is the main technical tool in the local study of schemes.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
Overview
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry.
In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
Connections with algebraic geometry
Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1677671318/643

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