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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 3rd 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## Terminology There are two inequivalent definitions of [[Fréchet spaces]] found in the literature. The original definition due to [[Stefan Banach]] defines [[Fréchet spaces]] as [[metrizable topological space|metrizable]] [[complete space|complete]] [[topological vector spaces]]. Later [[Bourbaki]] (Topological vector spaces, Section II.4.1) added the condition of [[locally convex vector space|local convexity]]. However, many authors continue to use the original definition due to Banach. The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion. The [[nLab]] uses “F-space” to refer to the non-locally convex notion and “[[Fréchet space]]” to refer to the locally convex notion. <a href="https://ncatlab.org/nlab/revision/diff/F-space/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/F-space/9">v9</a>, <a href="https://ncatlab.org/nlab/show/F-space">current</a>

   Added:

   Terminology

   There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces.

   Later Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. However, many authors continue to use the original definition due to Banach.

   The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion.

   The nLab uses “F-space” to refer to the non-locally convex notion and “Fréchet space” to refer to the locally convex notion.

   diff, v9, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 3rd 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: ## References * [[Nigel Kalton]], _Quasi-Banach spaces_, Handbook of the geometry of Banach spaces, Volume 2, 1099–1130. North-Holland, Amsterdam, 2003. ISBN: 0-444-51305-1. <a href="https://ncatlab.org/nlab/revision/diff/F-space/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/F-space/9">v9</a>, <a href="https://ncatlab.org/nlab/show/F-space">current</a>

   Added:

   References

   • Nigel Kalton, Quasi-Banach spaces, Handbook of the geometry of Banach spaces, Volume 2, 1099–1130. North-Holland, Amsterdam, 2003. ISBN: 0-444-51305-1.

   diff, v9, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 4th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: * N. Kalton, N. T. Peck, James W. Roberts, _An F-space sampler_, Cambridge University Press (1984), London Mathematical Society Lecture Notes 89, Cambridge. ISBN: 9780511662447, [DOI](https://doi.org/10.1017/CBO9780511662447). * Norbert Adasch, Bruno Ernst, Dieter Keim, _Topological Vector Spaces: The Theory Without Convexity Conditions_, Lecture Notes in Mathematics 639 (1978), Springer. ISBN 978-3-540-08662-8. <a href="https://ncatlab.org/nlab/revision/diff/F-space/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/F-space/10">v10</a>, <a href="https://ncatlab.org/nlab/show/F-space">current</a>

   Added:

   • N. Kalton, N. T. Peck, James W. Roberts, An F-space sampler, Cambridge University Press (1984), London Mathematical Society Lecture Notes 89, Cambridge. ISBN: 9780511662447, DOI.

   • Norbert Adasch, Bruno Ernst, Dieter Keim, Topological Vector Spaces: The Theory Without Convexity Conditions, Lecture Notes in Mathematics 639 (1978), Springer. ISBN 978-3-540-08662-8.

   diff, v10, current