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continuous metric space valued function on compact metric space is uniformly continuous
…
…
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$(\esh \dashv \flat \dashv \sharp )$
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$(\Re \dashv \Im \dashv \&)$
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Since every ordered Artinian local ring $R$ has characteristic zero, the positive integers $\mathbb{Z}_+$ are a subset of $R$, with injection $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered Artinian local ring is an ordered Artinian local ring which satisfies the archimedean property: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.
Archimedean ordered Artinian local rings are important because they are the ordered local rings with nilpotent infinitesimals but no infinite elements or invertible infinitesimals, and thus play an important role in synthetic approaches to analysis and differential geometry.
Archimedean ordered fields are the Archimedean ordered Artinian local rings in which the nilradical is the trivial ideal, or equivalently, in which every non-invertible element is equal to zero.
Suppose that $K$ is an Archimedean ordered field and $A$ is an Archimedean ordered Artinian local $K$-algebra. Since $A$ is a local ring, the quotient of $A$ by its ideal of non-invertible elements $D$ is the residue field $K$ itself, and the canonical function used in defining the quotient is the function $\Re:A \to K$ which takes a number $a \in A$ to its purely real component $\Re(a) \in K$. Since $A$ is an ordered $K$-algebra, there is a strictly monotone ring homomorphism $h:K \to A$. An element $a \in A$ is purely real if $h(\Re(a)) = a$, and an element $a \in A$ is purely infinitesimal if it is in the fiber of $\Re$ at $0 \in K$. Zero is the only element in $A$ which is both purely real and purely infinitesimal.
Now, suppose that the Archimedean ordered field $K$ has lattice structure $\min:K \times K \to K$ and $\max:K \times K \to K$. Then the Archimedean ordered local $K$-algebra $A$ has a prelattice structure given by functions $(-)\wedge(-):A \times A \to A$ and $(-)\vee(-):A \times A \to A$, defined by
and
The distance function on $A$ is given by the function $\rho:A \times A \to K$, defined as
and the absolute value on $A$ is given by the function $\vert-\vert:A \times A \to K$, defined as
The distance function and absolute value are pseudometrics and multiplicative seminorms, because every Archimedean ordered field $K$ embeds into the real numbers $\mathbb{R}$, and since $\min(a, b) \leq \max(a, b)$, the pseudometric and seminorm are always non-negative. This also implies that in every Archimedean ordered field with lattice structure, the pseudometric defined above is a metric.
Suppose that $K$ is an Archimedean ordered field with lattice structure, and $A$ is an Archimedean ordered Artinian local $K$-algebra. Then continuous functions, differentiable functions, and smooth functions are each definable on $K$ using the algebraic, order, and metric structure on $K$.
The ring homomorphism $h:K \to A$ preserves smooth functions: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$, then for every smooth function $f \in C^\omega(K)$, there is a function $f_A:A \to A$ such that for all elements $x \in K$, $f_A(h(x)) = h(f(x))$ and
If we restrict to Archimedean ordered Artinian local $K$-algebras $A$ where every element of the nilradical $D$ is a nilsquare element, where for all $\epsilon \in D$, $\epsilon^2 = 0$, then the ring homomorphism $h:K \to A$ preserves differentiable functions; for every differentiable function $f:K \to K$ with given derivative $f':K \to K$, there is a function $f_A:A \to A$ such that for all elements $x \in K$ and nilpotent elements $\epsilon \in D$, $f_A(h(x)) = h(f(x))$ and
In particular, every polynomial function $p:K \to K$ lifts to a polynomial function $p_A:A \to A$.
Alternatively, one could use this property to define differentiable and smooth functions in $K$, such as the exponential function, natural logarithm, sine function, and cosine function.
One could also work with partial functions instead. Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A partial function $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.
A function $f:I \to \mathbb{R}$ is smooth at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $(D^0 j)(a) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$
A function $f:I \to \mathbb{R}$ is smooth at an element $a \in I$ if it is smooth at the singleton subset $\{a\}$, and a function $f:I \to \mathbb{R}$ is smooth if it is smooth at the improper subset of $I$.
A function $f:I \to \mathbb{R}$ is differentiable at a subset $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d f}{d x}:S \to \mathbb{R}$ such that for all Archimedean ordered Artinian local $K$-algebras $A$ with ring homomorphism $h:K \to A$ such that for all $\epsilon \in D$, $\epsilon^2 = 0$, for all nilpotent elements $\epsilon \in D$,
A function $f:I \to \mathbb{R}$ is differentiable at an element $a \in I$ if it is differentiable at the singleton subset $\{a\}$, and a function $f:I \to \mathbb{R}$ is differentiable if it is differentiable at the improper subset of $I$.
Now, assume that $K$ is an Euclidean field as well, in addition to being an Archimedean ordered field. While $K$ has a principal square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$, not every Archimedean ordered local $K$-algebra $A$ has a principal square root function $\sqrt{-}:[0, \infty) \to [0, \infty)$, because purely infinitesimal elements in $A$ are not guaranteed to have square roots. An Archimedean ordered Artinian local $K$-algebra is Euclidean if every nilpotent element has a square root.
However, given an Archimedean ordered Artinian local $K$-algebra $A$, every rank $n$ $A$-module $V$ with basis $v:\mathrm{Fin}(n) \to V$ has a Euclidean pseudometric $\rho_V:V \times V \to K$, given by
for module elements $a \in V$ and $b \in V$ and scalars $a_i \in A$ and $b_i \in A$ for index $i \in \mathrm{Fin}(n)$, where
If $A$ is an ordered field, then the Euclidean pseudometric on $V$ is a metric.
Last revised on January 13, 2023 at 05:38:36. See the history of this page for a list of all contributions to it.