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Questions tagged [frechet-space]

17 questions
4
votes
2 answers

Motivation behind locally convex spaces, seminorms, and Frechet spaces

I am looking for some motivation behind the definition of locally convex spaces, seminorms, and Frechet spaces. Since all three concepts are related I have grouped them as one question. I am familiar with the technical definitions but I don't see…
CBBAM
  • 7,149
3
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0 answers

Spectral theory for Frechet algebra?

In most textbooks discussing spectral theory of operators, they focus on a Banach algebra of operators due to the power that completeness provides. Frechet spaces are complete metric spaces too, so is there a well-developed spectral theory for a…
Ishan Deo
  • 3,886
3
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Strict inductive limit of sequentially complete locally convex spaces

If $\{A_n\}$ is a strict inductive sequence of Hausdorff locally convex spaces (meaning each of the connecting maps $\iota_{n,n+1}:A_n \rightarrow A_{n+1}$ is a topological inclusion) with each $A_n$ being sequentially complete. Then is…
mathable
  • 484
3
votes
1 answer

Comparison between the summability of fourier coefficients as smooth and $L^2$ function

Let $\mathbb{T} = [0,2\pi]$. It is is well-known that $ L^2(\mathbb{T})$ and $ l_2(\mathbb{Z})$ are isometrically isomorphic via the Fourier transform and the Fourier series is its inverse. Parseval's identity states that $\lVert f \rVert_2^2 =…
mixotrov
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3
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Is there an abstract reason why alot of calculus tools that exist on Banach spaces fail in Frechet spaces?

As far as I can tell, which doesnt have to mean much, Frechet spaces are basicly the "closest possible" generalisation of Banach spaces, since every Frechet space is a sequential limit of Banach spaces in the category of locally convex topological…
Benjamin Rogoll
  • 43
3
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0 answers

Perturbation of semi-Fredholm operators in Frèchet spaces

It is well known that the index is a continuous function on the set of Semi-Fredholm operators on a Banach space, and even on quasi-Banach spaces. The result is unfortunately false in general Fréchet spaces, as exemplified below. I am interested in…
Pelota
  • 1,362
3
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1 answer

Finding a set of "graded" seminorms on the Schwartz space

For any fixed $N \in \mathbb{N}$, let $\mathcal{S}(\mathbb{R}^N)$ be the Schwartz space. Then, it is well-known that $\mathcal{S}(\mathbb{R}^N)$ is a Fréchet space with the seminorms: \begin{equation} \lVert f \rVert_{n, \alpha} := \sup_{ x \in…
Keith
  • 8,359
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What minimal properties of a TVS ensure that the closure of an open ball is the closed ball?

Let $X$ be a metrizable topological vector space (TVS) over $\mathbb R\text{ or }\mathbb C$, and consider the following two properties that $X$ might have. Local convexity Completeness (as a TVS, which implies completeness in any invariant…
WillG
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2
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0 answers

why quojections are prequojections

While reading about quojections and prequojections in the book "Advances in the Theory of Fréchet Spaces," I'm having trouble understanding why every quojection is necessarily a prequojection. Could someone clarify the reasoning behind this? by the…
Assalami Med
  • 21
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2 answers

Condition for showing families of seminorms generate same topology

There is a statement about locally convex spaces in Reed & Simon, Methods of Modern Mathematical Physics (Vol I, Section V.1) that is given without a proof. The statement is: Given two families of seminorms $\{\rho_\alpha\}_{\alpha \in A}$ and…
Mark
  • 134
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Is the dual of the space of semi-regular distributions complete

Consider $C^\infty(X)$, equipped with its Fr'echet structure, and $\mathcal D'(Y)$, equipped with the strong topology, where $X,Y$ are compact manifolds. Is the strong dual of $C^\infty(X)\hat\otimes \mathcal D'(Y)$ a complete topological vector…
Or Kedar
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Is the Fréchet quotient space given by the induced seminorms?

We admit the following definition of Fréchet space. The definition of Fréchet space: Let $X$ be a linear space and let $\{p_i:X\to\mathbb R^+\}_{i\in \mathbb N}$ be a countable family of seminorms such that $p_i(x)\ne 0$ for some $i\in\mathbb N$…
Chiquita
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1 answer

Is there a preferred topology making the general linear group of a Frechet space into a topological group?

Is there a preferred topology on the isomorphisms of a Frechet space giving them the structure of a topological group? I was playing with the metric and strong operator topology on the general linear group of Schwartz space…
gordta_chichrron
  • 571
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Continuous maps between Frechet spaces (related to definition of Fourier multiplier )

I'm trying to understand a proof of the following simple statement: Let $A:C^\infty(\mathbb{S}^1,\mathbb{C})\to C^\infty(\mathbb{S}^1,\mathbb{C})$ be a continuous linear operator on the Frechet space $C^\infty(\mathbb{S}^1,\mathbb{C})$. If $A$…
Stuck
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1 answer

Complemented range operators on Fréchet spaces

Let $F$ be a Fréchet space and $A:F\rightarrow F$ a continuous linear operator such that $A\left( F\right) $ is closed and there is some closed subspace $E\subset F$ such that $F=A\left( F\right) \oplus E$. Is there some neighborhood $V$ of $0$ such…
Djalal Ounadjela
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