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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 what would lead one to defining them.

What is so special about locally convex spaces that we wish to focus are analysis only on them?

I am aware that seminorms are generalizations of norms but with the condition $\|x \| = 0 \implies x = 0$ dropped. But why do we care about such objects?

The idea of angles and inner products would naturally lead one to define a Hilbert space and similarly length would lead one to define Banach spaces. Is there any similar idea that a Frechet space captures?

CBBAM
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  • (not posting as an answer because not comprehensive) I would point out that for open $\Omega \subseteq \mathbb R^d$ and $k \in \mathbb Z_{\ge 1} \cup {\infty}$, $C^k(\Omega)$ is a Frechet space that is not normable. Distributions can be understood as the continuous linear functionals on $C_0^\infty(\Omega) \subseteq C^\infty(\Omega)$, understood as a locally convex space, and so on. The choice of seminorms for $C^k$ is $p_{\alpha, K}(u) = \sup_K |\partial^\alpha u|$ for each compact $K \subseteq \Omega$ and multi-index $|\alpha| \le k$. – George Coote Jan 02 '24 at 11:20
  • oh, and to give some sense to this, the topology on $C^k$ then becomes that of uniform convergence on compacts. So $u_n \to u$ in this topology iff $\sup_K |\partial^\alpha u_n - \partial^\alpha u| \to 0$ for each compact $K$ and suitable $\alpha$ – George Coote Jan 02 '24 at 11:31
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    You should probably read all that from the lion's paws. See Dieudonne, J. History of Functional Analysis Chapter VIII. Section 2. – Mittens Jan 02 '24 at 17:27
  • Spaces of $C^\infty$ functions are never Banach spaces but are Frechet spaces. So if for example you want to prove the existence of a smooth solution to a nonlinear PDE, it is sometimes best to use Frechet spaces. This is the case if you want to use the Nash implicit function theorem. – Deane Jan 02 '24 at 18:05
  • @MarianoSuárez-Álvarez I think the existence of motivating examples answers my question, but I was wondering if there was, say, a geometric motivation as we have in Hilbert or Banach spaces – CBBAM Jan 02 '24 at 22:17
  • @Deane I see, so the definition is as it is because it captures a lot of frequently occurring examples? Is there any geometric idea that Frechet captures like Banach or Hilbert spaces do? – CBBAM Jan 02 '24 at 22:18
  • @MarianoSuárez-Álvarez But is there any overarching theme that Frechet spaces or seminorms capture? For example Hilbert spaces and inner products are defined the way they are, as you mention, based on examples that all involve angles. But all the examples have the same theme of angles. I was wondering if seminorms or Frechet spaces have any similar overarching theme. – CBBAM Jan 02 '24 at 22:47
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    Finite dimensional Hilbert and Banach spaces can be viewed as geometric space, but I have never viewed the infinite dimensional versions as geometric spaces. In applications, these spaces are usually spaces of functions, where the linear and topological structures are quite useful. I don't have any geometric intuition for these spaces. I don't think of these spaces as geometric spaces. A Frechet space is nontrivial only if it is infinite dimensional. Again, for me it is useful only as a space of functions. – Deane Jan 02 '24 at 22:49
  • @Deane Thank you for your comment. – CBBAM Jan 02 '24 at 23:03
  • @MarianoSuárez-Álvarez Thank you, that explanation of seminorms capturing only part of a thing is very helpful. – CBBAM Jan 02 '24 at 23:03

2 Answers2

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Why Locally Convex Topological Vector Spaces?

Some reasons why the attention is focused on locally convex (Hausdorff) spaces:

  • The topological vector space topology generated by a family of semi norms is locally convex (meaning that such a topology arises quiet naturally, because semi norms arise naturally in the study of function spaces and operator spaces)
  • Many interesting topological vector space topologies generated by a family of semi norms are neither normable nor metrizable.
  • The theorem of Hahn-Banach remains true for locally convex spaces (Hahn Banach gets brutally violated even by metrizable topological vector spaces, this is one of the reasons why we dont focus our attention on those)

Some of the motivation for the study of locally convex spaces:

  • The weak topology on a Hilbert space is not metrizable (and therefore also not normable). The first to show this was von Neumann, who shortly after was the first to define a locally convex topological vector space.
  • Schwartz, Dieudonne, Grothendieck and others were motivated to develop the theory of locally convex spaces as we know it today by the study of function spaces and distributions

Example: Locally Convex Topology Of Pointwise Convergence

In the context of function spaces seminorms and locally convex topological vector spaces arise naturally, as the following example shows:

Let $S$ be any set and let $F$ be the set of all functions $S \to \mathbb{C}$ with the obvious (pointwise) vector space operations. Then for each $s \in S$ we can define a semi-norm $p_s: F\to [0,\infty)$ by $$p_s(f) = | f(s) | \quad \forall f \in F.$$

The semi norm $p_s$ does not give us the "size" (or length) of the function, but it gives us "partial information" about the size of $f$ (namely the size of $f$ at $s$). This is how semi-norms are usually defined in this context: They give us the "size" of some aspect of the function.

Together all the semi norms $(p_s)_{s \in S}$ give us a complete "picture" of the function. Namely if $f\neq 0$, then there exist a $s \in S$ with $p_s(f) \neq0$. This property will translate to the fact that the topological vector space topology generated by this family of semi-norms is point separating.

Now endow $F$ with the topological vector space topology $\tau$ generated by the family of semi-norms $(p_s)_{s \in S}$. Then (by a general result) a sequence $(f_n)_{n \in \mathbb{N}}$ in $F$ (or more generally a net) converges to $f\in F$ with respect to $\tau$ if and only if $$ \lim_{n \to \infty} p_s(f-f_n)= 0 \quad \forall s \in S. $$ In other words $f_n \overset{\tau}{\to} f$, if and only if $f_n(s) \overset{| \cdot| }{\to} f(s)$ for all $s \in S$.

Therefore the elementary idea of pointwise convergence (that can be found in even the most basic analysis book) is captured with this topology, which is clearly a very usefull thing.

Why Frechet Spaces?

The (or at least part of the) reason why Frechet spaces are interesting is that

  • a few major theorems in Banach space functional analysis hold for Frechet spaces as well (some are listed here)
  • Some interesting locally convex spaces (that are not normable) are Frechet spaces (for example the Schwartz space or the space described in the comment by George C)

I dont think that there is any deeper motivation than the above (although i might be wrong of course). I think they are used, because someone realised that being a Frechet space is enough to prove some of the major Banach space theory results and that some interesting spaces are Frechet spaces.

jd27
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  • @MarianoSuárez-Álvarez But that does not explain why we should care about locally convex topological vector spaces instead of say topological vector spaces or any other way to generalize banach spaces. – jd27 Jan 02 '24 at 21:34
  • Thank you, this was extremely insightful! – CBBAM Jan 02 '24 at 22:25
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In Dieudonne's "History of Functional Analysis", Chapter VIII is on locally convex spaces and distributions, and Section 2 discusses the development of locally convex spaces. In the 1st paragraph he writes

the various notions belonging to what we now call the general theory of topological vector spaces made their appearance in a rather random way and were not the subject of a systematic treatment until 1950.

In 1906 Mazur and Orlicz defined Frechet spaces (called by them "spaces of type $B_0$") and in 1935, von Neumann defined locally convex spaces (called by him "convex spaces").

Dieudonne particularly emphasizes the work of Grothedieck on the tensor product of locally convex spaces, calling it

the greatest progress in functional analysis after the work of Banach.

In the 1950s, Grothendieck introduced nuclear spaces in his work on topological tensor products. Nuclear spaces are locally convex and generalize Euclidean spaces in a quite different way than Banach spaces: a nuclear space is a Banach space if and only if it is finite-dimensional. So perhaps it was the introduction of nuclear spaces that brought attention to the need for a systematic development of locally convex spaces and more generally topological vector spaces.

KCd
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  • Thank you. Much of the reason I am interested in this area of functional analysis is to build up to nuclear spaces. How do these generalize Euclidean space? And do you have any references you can recommend to learn about nuclear spaces? – CBBAM Jan 02 '24 at 22:28
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    @CBBAM See https://mathoverflow.net/questions/442204/nuclear-spaces-and-intuition-behind-their-topology both the answers and comments. – KCd Jan 03 '24 at 00:02
  • re: nuclear spaces behaving like finite-dimensional ones, this is particularly relevant in the case of multilinear algebra, e.g., the Schawrtz Kernel Theorem see https://math.stackexchange.com/questions/2623515/schwartz-kernel-theorem-and-dual-topologies/2647815#2647815 and in the case of probability theory where Bochner's Theorem and the Levy Continuity Theorem have formulations which are as clean and simple as in the finite-dimensional situation, see https://mathoverflow.net/questions/384124/bochner-minlos-for-moment-generating-functions/384168#384168 ... – Abdelmalek Abdesselam Jan 06 '24 at 15:53
  • ...this is especially useful in quantum field theory which deals with probability measures on spaces of distributions like $\mathscr{S}'$. – Abdelmalek Abdesselam Jan 06 '24 at 15:54
  • @AbdelmalekAbdesselam Thank you for your comment. Quantum field theory is actually the reason I am interested in nuclear spaces. I tried reading some of the literature on rigorous QFT but I noticed I need to learn more about nuclear spaces/operators and probability theory on distribution spaces before continuing. For this I've been reading through the fourth volume of Gelfand and Vilenkin's series on generalized functions. Is there any other resource you recommend or is their treatment of these subjects enough to read, for example, Glimm & Jaffe's book? – CBBAM Jan 11 '24 at 02:24
  • Unfortunately, that's a lot of wasted effort, since you have taken the very long way around when there is much more direct route: learn the isomorphism with spaces of sequences, prove the Bochner-Minlos theorem for sequence spaces, and then transfer it to $S'$. This is what I have been explaining in my previous answers. For learning about probability theory on spaces of distributions in 2024, the books by GJ and GV are not the best source. – Abdelmalek Abdesselam Jan 11 '24 at 15:18
  • If you email me I can send you lecture notes on distributions where the sequence space isomorphism is done in detail. For Bochner-Minlos and Levy Continuity see https://arxiv.org/abs/1706.09326 – Abdelmalek Abdesselam Jan 11 '24 at 15:24
  • @AbdelmalekAbdesselam Thank you very much! I will send you an email. – CBBAM Jan 11 '24 at 21:08
  • @AbdelmalekAbdesselam I am having some issues with my email. May we discuss this further in a stackexchange chat? – CBBAM Jan 14 '24 at 08:30
  • @CBBAM: I don't correspond by email with pseudonyms. In any case, I just resurrected my webpage and the notes I talked about are posted here https://mabdesselam.github.io/MATH7320Fall2017.html The sequence space stuff is in Chapter 4. – Abdelmalek Abdesselam Jan 15 '24 at 01:43
  • @AbdelmalekAbdesselam Thank you! – CBBAM Jan 15 '24 at 05:10