symmetric monoidal (∞,1)-category of spectra
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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A real polynomial function is a polynomial function in the real numbers, a function $f:\mathbb{R} \to \mathbb{R}$ such that
$f$ is in the image of the function $j:\mathbb{R}^* \to (\mathbb{R} \to \mathbb{R})$ from the free monoid $\mathbb{R}^*$ on $\mathbb{R}$, i.e. the set of lists of real numbers, to the function algebra $\mathbb{R} \to \mathbb{R}$, such that
$f$ is in the image of the canonical ring homomorphism $i:\mathbb{R}[x] \to (\mathbb{R} \to \mathbb{R})$ from the real polynomial ring in one indeterminant $\mathbb{R}[x]$ to the function algebra $\mathbb{R} \to \mathbb{R}$, which takes constant polynomials in $\mathbb{R}[x]$ to constant functions in $\mathbb{R} \to \mathbb{R}$ and the indeterminant $x$ in $\mathbb{R}[x]$ to the identity function $\mathrm{id}_\mathbb{R}$ in $\mathbb{R} \to \mathbb{R}$
There exists a natural number $n$ such that the $n$-th order derivative of $f$ is equal to the zero function:
A real polynomial function is a function $f:\mathbb{R} \to \mathbb{R}$ with a natural number $n \in \mathbb{N}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ which satisfy one of these conditions:
for all $x \in R$,
where $x^i$ is the $i$-th power function for multiplication.
$f$ is a solution to the $n$-th order linear homogeneous ordinary differential equation
with initial conditions
for each natural number $i:[0, n)_\mathbb{N}$
Given a non-zero real polynomial function $f:\mathbb{R} \to \mathbb{R}$, the degree of $f$ is the maximum natural number $n$ where the $n$-th derivative of $f$ is not the zero-function:
The degree of a real polynomial function, with additional data of a natural number $n \in \mathbb{N}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$, is defined as the maximum natural number $i \lt n$ such that $\vert a_i \vert \gt 0$.
In constructive mathematics, there exist real polynomial functions for which one cannot prove that a particular natural number $i \lt n$ is the degree of the real polynomial function: i.e. the function sub-$\mathbb{R}$-algebra of real polynomial functions is not a Euclidean domain.
Given a real polynomial function $f:\mathbb{R} \to \mathbb{R}$, $f$ is a pointwise continuous function with respect to its metric topology defined through the absolute value function, subtraction, and its strict total order.
Given a real polynomial function $f:\mathbb{R} \to \mathbb{R}$, $f$ is a pointwise differentiable function with respect to its metric topology defined through the absolute value function, subtraction, and its strict total order.
Every real polynomial function is a smooth function. This could be shown coinductively: A smooth function is a pointwise differentiable function whose derivative is also smooth, and thus the $n$-th derivative of a smooth function is a smooth function. The zero function $f(x) = 0$ is a smooth function, every real polynomial function is a pointwise differentiable function and the $n$-th derivative of a polynomial function, with a natural number $n \in \mathbb{R}$ and a list of length $n$ of real numbers $a:[0, n)_\mathbb{N} \to \mathbb{R}$ such that one of the two conditions above is satisfied, is the zero function. Thus, every polynomial function is a smooth function.
The proof of pointwise continuity using epsilontic analysis is spelled out for instance in
See also:
Last revised on December 24, 2023 at 22:11:32. See the history of this page for a list of all contributions to it.