By definition, a locally convex space X is a Frechet space, if it has two properties

1) X is metrizable, and

2) X is complete.

For each of these properties there are many easy examples of spaces that do not possess it. At first, for metrizability: in order to be metrizable X must have a countable base of neighborhoods of zero. The following spaces do not have this property: R[0,1] , that is the space of all real-valued functions on [0, 1] equipped with the topology of pointwise convergence (product topology); any infinite-dimensional Banach space equipped with the weak topology; any infinite-dimensional linear space over reals equipped with the strongest locally convex topology.

For completeness. If a subspace X of Hausdorff topological vector space E is complete, then it is closed. So any non-closed subspace in a locally convex Hausdorff space gives you an example of a non-complete space.

A-1 LF-spaces give some wanted examples

Answer 1, posted on October 21, 2019.

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LF-spaces give examples of locally convex topological vector spaces

that are not Fréchet spaces:

https://en.wikipedia.org/wiki/LF-space

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With best regards, Jean-Claude

By definition, a locally convex space X is a Frechet space, if it has two properties

1) X is metrizable, and

2) X is complete.

For each of these properties there are many easy examples of spaces that do not possess it. At first, for metrizability: in order to be metrizable X must have a countable base of neighborhoods of zero. The following spaces do not have this property: R[0,1] , that is the space of all real-valued functions on [0, 1] equipped with the topology of pointwise convergence (product topology); any infinite-dimensional Banach space equipped with the weak topology; any infinite-dimensional linear space over reals equipped with the strongest locally convex topology.

For completeness. If a subspace X of Hausdorff topological vector space E is complete, then it is closed. So any non-closed subspace in a locally convex Hausdorff space gives you an example of a non-complete space.

A-3. Four related books

Answer 3, posted on October 22, 2019.

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Vladimir Kadets has published the following books:

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A course in functional analysis and measure theory

Universitext

Vladimir Kadets

Translated from Russian into English by Iacob, A.

Original edition in Russian: 2006

DOI: 10.1007/978-3-319-92004-7

ISBN-13: 9783319920030

539 + xxii pages, Springer, 2018

https://www.springer.com/us/book/9783319920030

https://www.researchgate.net/publication/266952376_A_course_in_functional_analysis_and_measure_theory_Kurs_funkcionalnogo_analizu_ta_teorii_miri_Ukrainian_Transl_from_the_Russian_A_course_in_functional_analysis

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A course in functional analysis

Textbook for students of mechanics and mathematics

Vladimir Kadets

Publisher: Khar'kovskij Natsional'nyj Universitet Im. V. N. Karazina

ISBN: 966-623-199-9/hbk

December 2005

https://www.researchgate.net/publication/235223762_A_course_in_functional_analysis_Textbook_for_students_of_mechanics_and_mathematics

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Series in Banach spaces

Conditional and unconditional convergence

M.I. Kadets and Vladimir Kadets

Publisher: Birkhauser

ISBN: 3-7643-5401-1

December 1996

https://www.researchgate.net/publication/235219251_Series_in_Banach_spaces_Conditional_and_unconditional_convergence

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Rearrangements of series in Banach spaces.

Transl. from the Russian by Harold H. McFaden

Vladimir Kadets and M.I. Kadets

Publisher: American Mathematical Society

October 1991

https://www.researchgate.net/publication/265571850_Rearrangements_of_series_in_Banach_spaces_Transl_from_the_Russian_by_Harold_H_McFaden

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It is very likely that many of the concepts

he talks about in answer 2 are carefully

explained in his books.

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With best regards, Jean-Claude

A-4. Conditions for an LF-space to be a Fréchet space?

Answer 4, posted on October 23, 2019.

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Is there a publication that gives conditions

for an LF-space to be or not to be a Fréchet space?

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By definition, an LF-space is a topological vector space

that is a locally convex inductive limit

of a countable inductive system of Fréchet spaces.

https://en.wikipedia.org/wiki/LF-space

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So my question is equivalent to:

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“Is there a publication that gives conditions

for such a system of Fréchet spaces.

to have an inductive limit that is or that is not a Fréchet space?”

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With best regards, Jean-Claude