Why is $\P_d$ closed in $C([a,b])$?

Recall that $\P_d$ is the set of all real polynomials of degree at most $d$, and we consider the space $C([a,b])$ equipped with the $\sup$ norm. The claim is that $\P_d$ is a closed subset of $C([a,b])$. This is actually quite tricky to show—at least, I don’t know of a simple argument. Here is what…

Math 4A03: Proof of the last lemma on connectedness

Let $(M,d)$ be a metric space; we consider $M^2$ with the corresponding product metric. The goal of this post is to give a proof of the last lemma on these slides: Lemma Let $A,B \subseteq M$ be connected. Then $A \times B$ is connected. The proof uses the exercises preceding this lemma given on the…

Analytic continuation of $\log$-$\exp$-analytic germs

Our preprint on the analytic continuation of germs at $+\infty$ of unary functions definable in $\Ranexp$ is now on the ArXiv. Here is its introduction: The o-minimal structure $\Ranexp$, see van den Dries and Miller or van den Dries, Macintyre and Marker, is one of the most important regarding applications, because it defines all elementary…

Ilyashenko algebras: putting it all together

Let $f = (f_0, \dots, f_k)$ be such that each $f_i \in \H$ is infinitely increasing and $f_0 \gt \cdots \gt f_k$. To see what it takes to generalize our construction of the Ilyashenko algebra $(\F,L,T)$ to more general monomials $f$, recall the construction in the following schematic: $$ \begin{matrix} \RR & \xrightarrow{\text{(UP)}} & \begin{bmatrix}…

Ilyashenko algebras based on definable monomials: the construction (inductive step)

Let $M \subseteq \H^{>0}$ be a pure scale on standard power domains. In this post, I gave the base step of the construction of a qaa field $(\F,L,T)$ as claimed here. The goal of this post is to finish this construction. Step 0.5: apply a $\log$-shift to the qaa field $(\F_0,L_0,T_0)$, that is, set $$\F’_1…

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