Motivation behind locally convex spaces, seminorms, and Frechet spaces

Question

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?

Answer 1

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

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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.

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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.

Answer 2

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.