Questions tagged [rings-of-functions]
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7 questions
3
votes
1
answer
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What are the conditions for a function to be expressed as a sum of multiplicatively separable functions?
Let $f$ be a smooth real-valued function defined on a product domain $ U\times V $ of $\mathbb{R}^{n}\times \mathbb{R}^{m}$. I am interested in the conditions under which f can be written as a finite ...
- 103
4
votes
0
answers
219
views
Can an ideal in the ring of holomorphic functions on the complex plane be non-finitely generated?
Let $( I )$ be an ideal in the ring $( R )$ of all holomorphic functions of a single complex variable on the complex plane. I am interested in understanding whether it is possible for $( I )$ to be ...
- 93
3
votes
1
answer
259
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Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 845
11
votes
1
answer
512
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Maximal ideals of the ring $\mathbb C \{T\}$
Consider the Banach $\mathbb C$-algebra
$$
\mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace
$$
With the norm given by $\| \sum a_i T^i\| = \sum |...
45
votes
4
answers
4k
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Do rings of smooth functions differ from rings of continuous functions?
Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?
- 8,050
4
votes
2
answers
701
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Real spectrum of ring of continuous semialgebraic functions
Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$_r (R[...
- 1,332
27
votes
2
answers
9k
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Maximal ideals in the ring of continuous real-valued functions on ℝ
For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
- 6,844