Wikipedia:Articles for deletion/Differentiable vector–valued functions from Euclidean space

Differentiable vector–valued functions from Euclidean space

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Differentiable vector–valued functions from Euclidean space (edit | talk | history | protect | delete | links | watch | logs | views) – (View log | edits since nomination)

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Differentiable vector–valued functions from Euclidean space has interrelated issues. I'm not able to find other sources than the sole one that the article cites (F. Trèves' book on topological vector spaces). I think inasmuch as it is different from just, multivariable differential calculus, it is not a notable topic—in that sense, it may be seen a content fork, where the page is about an obscure TVS approach to a well-known topic that probably doesn't merit coverage on the article about the latter. It is also written in WP:NOTTEXTBOOK-like style, quite closesly paraphrasing Trèves. For example, the portion starting at Differentiable vector–valued functions from Euclidean space#Space of Ck functions corresponds tightly to the portion of Trèves starting at Notation 40.1; see an example of this below:

Article:

Suppose $\Omega _{1}\subseteq \Omega _{2}\subseteq \cdots$ is a sequence of relatively compact open subsets of $\Omega$ whose union is $\Omega$ and that satisfy ${\overline {\Omega _{i}}}\subseteq \Omega _{i+1}$ for all $i.$ Suppose that $\left(V_{\alpha }\right)_{\alpha \in A}$ is a basis of neighborhoods of the origin in $Y.$ Then for any integer $\ell <k+1,$ the sets: $${\mathcal {U}}_{i,\ell ,\alpha }:=\left\{f\in C^{k}(\Omega ;Y):\left(\partial /\partial p\right)^{q}f(p)\in U_{\alpha }{\text{ for all }}p\in \Omega _{i}{\text{ and all }}q\in \mathbb {N} ^{n},|q|\leq \ell \right\}$$ form a basis of neighborhoods of the origin for $C^{k}(\Omega ;Y)$ as $i,$ $\ell ,$ and $\alpha \in A$ vary in all possible ways.

Trèves:

Consider a sequence $\Omega _{1}\subset \Omega _{2}\subset \dots \subset \Omega _{j}\subset \dots$ of relatively compact open subsets of $Y$ whose union is equal to $Y$, an arbitrary integer $l<k+1$, a basis of neighborhoods of zero in $E$, [namely] $\{U_{\alpha }\}$. As $j,l$ and $\alpha$ vary in all possible ways, the subsets of $C^{k}(Y;E)$, $${\mathcal {U}}_{j,l,\alpha }=\left\{{\underline {f}}:\left(\partial /\partial y\right)^{q}{\underline {f}}(y)\in U_{\alpha }{\text{ for all }}y\in \Omega _{j}{\text{ and all }}q\in \mathbb {N} ^{n},|q|\leq l\right\},$$ form a basis of neighborhoods of zero for the $C^{k}$ topology.

ByVarying | talk 02:49, 24 March 2025 (UTC)

• Note: This discussion has been included in the list of Mathematics-related deletion discussions. CAPTAIN RAJU^(T) 03:16, 24 March 2025 (UTC)

• Keep: I do not agree with your assertion about a lack of noteworthiness. Also, the topic is substantially different from multivariate calculus. Topological vector spaces are certainly noteworthy, and since differentiability is a corner stone in analysis, it is clear, that differentiable functions with values in such spaces are also noteworthy. The classical definition of differentiability is based on norms (see, for instance, Jean Dieudonné's textbook "Foundations of Modern Analysis"). This classical approach does not work for functions with values in topological vector spaces. So this article has very little to do with multivariate differential calculus. It is a keep. 51.154.152.231 (talk) 16:23, 30 March 2025 (UTC)

  When I say notable, I'm talking about WP:N. Being tangentially related to notable topics doesn't make something notable; substantial coverage in RS does. ByVarying | talk 00:58, 31 March 2025 (UTC)

Relisted to generate a more thorough discussion and clearer consensus.
Please add new comments below this notice. Thanks, ✗plicit 11:30, 31 March 2025 (UTC)

• Keep Some Advanced Calculus texts use the approach of maps between finite dimensional vector spaces. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:50, 1 April 2025 (UTC)
• Keep/Merge it seems like a content issue and the deletion is not need to solve such an issue. Perhaps the article should be merged into some other articles? —- Taku (talk) 10:32, 2 April 2025 (UTC)

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