# Tagged Questions

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### Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
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### Agreement of two topologies on a linear space

I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide. Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
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### Topological Generalization of Whitney's Extension Theorem

From Wikipedia: In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if $A$ ...
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### The pth power of a distance function is twice continuously differentiable, for $p>2$?

Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$. Is $\beta^p$, $p>2$ a twice continuously ...
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### Regular Borel Measures equivalent definition

Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
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### Riesz representation theorem for vector-valued fields

Let $Q$ be a locally compact Hausdorff space, and let $V$ be a topological vector space. Consider the space $X = C_0(Q, V)$ of $V$-valued fields which vanish at infinity. Let $X^*$ denote the dual ...
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### Extending a Hilbert space isometrically

Let $H$ be a Hilbert space, and let $X$ be a topological vector space. Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$? ...
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### Compactly generated Banach spaces

Suppose that $X$ is a Banach space (or more generally, Frechet space) such that $X$ is the closure of the span of a compact (in the original topology) subset $K$. Do we know anything "nice" about $X$, ...
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### When is a sequentially closed cone, closed?

The following question I also posed here, but still got no answer. Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What ...
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### Measures idempotent with respect to addition and multiplication.

Does there exist a probability finitely additive measure on $\mathbb N$ which is idempotent with respect to addition and multiplication simultaneously? It is known (due to Hindman) that there is no ...
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### Does this construction yield an injective hull ?

Let $K$ be an object of $\mathbf{CHaus}$, the category of compact Hausdorff spaces, and $K \xrightarrow{\ \ \sigma \ \ } K$ be an involutory morphism without fixed points. Define $C^{\sigma}(K)$ as ...
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### Topological properties of SpecMax(A)

We consider $A = C_{b}(X)$, the ring of continuous bounded functions on a completely regular space $X$. Let $\DeclareMathOperator{\SpecMax}{SpecMax} \SpecMax(A)$ be the set of maximal ideals of $A$ ...
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### a question about algebras between $C^{\ast}(X)$ and $C(X)$ [closed]

Let $X$ be a completely regular space, and $C(X)$ be the algebra of all real-valued continuous functions on a completely regular space $X$,and $C^{\ast}(X)$ is the subalgebra of bounded functions.and ...
### Is P(X) a connected set for a set X with a $\sigma$-algebra P(X) and a measure function m on it to [0,$\infty$] when P(X) is equiped with meter d, that for every A,B in P(X), $d(A,B)=m(A \Delta B)$?
An object $X$ of a given category is called projective if for each morphism $f : X \rightarrow Z$, and each epimorphism $g : Y \twoheadrightarrow Z$, there is a morphism $h : X \rightarrow Y$ such ...