There are two wolves inside you…

Consider a vector space $X$ over a field $\mathbb{K}$ (usually^{1 2 3} $\mathbb{R}$ or $\mathbb{C}$), endowed with a topology $\tau$ that is compatible with its vector space structure —that is, sum, and scalar multiplication are continuous operations.

In principle, we can consider the set of all linear maps from $X$ to the field $\mathbb{K}$. This set is called the algebraic dual of $X$ and is denoted by $X^* = \{ f:X\to\mathbb{K} \mid f \text{ is linear} \}.$ Note that this defines a vector space over $\mathbb{K}$, which is not endowed with any topology; it is, in principle, just an algebraic construct.

If we restrict ourselves to the set of continuous linear maps, we obtain the topological dual of $X$, denoted by $X' = \{ f:X\to\mathbb{K} \mid f \text{ is continuous w.r.t. } \tau, \; f \text{ is linear} \}.$ This is a vector space over $\mathbb{K}$, and even if in principle it is not endowed with a particular topology—thus, it is not a topological vector space—it can be endowed with different topologies⁴.

…and they seem to like each other…

At times, it is more convenient—as in theory of distributions—to put different spaces in duality with each other. In this case, one considers a (canonical) pairing $\langle \cdot, \cdot \rangle$ between two spaces $X$ and $Y$, which is a bilinear map in $X \times Y$. The main requirement is a separation condition (injectivity condition, non-degeneracy condition): if $\langle x, y \rangle = 0$ for all $y\in Y$, then $x=0$; similarly, if $\langle x, y \rangle = 0$ for all $x\in X$, then $y=0$.

…but let’s suppose they are nice…

The above were a glimpse of the duality theory, a very interesting topic treated with a lot of care in the literature.

Following this paper⁵, we restrict ourselves to the case of Banach spaces. A Banach space is a complete normed vector space, that is, a vector space $X$ over a field $\mathbb{K}$, endowed with a norm $\|\cdot\|$.

Despite the simplification, one can still consider a quite important question: what happens if we keep taking biduals of a Banach space $X$? Note that $X \hookrightarrow X''$ isometrically, where $X''$ is the bidual of $X$. This embedding is given naturally, namely,

$$\iota:X \to X'', \quad \iota(x)(f) = f(x), \text{ for all } f\in X'.$$

The article ⁵ studies this questions, and define the two natural categorical limits, making use of the standard constructions from TVS theory. Naturally, there are two sequences of biduals. Denote the $n$-th bidual by $X^{(n)}$, with $X^{(0)} = X$. Then, one can consider the two sequences:

1. The even biduals: $X^{(0)} \hookrightarrow X^{(2)} \hookrightarrow X^{(4)} \hookrightarrow \cdots$
2. The odd biduals: $X^{(1)} \twoheadleftarrow X^{(3)} \twoheadleftarrow X^{(5)} \twoheadleftarrow \cdots$

The main result of the paper is that both sequences converge to different limits, which we denote by

• $X^{(\infty, \mathsf{even})} = \varinjlim X^{(2n)}$, and
• $X^{(\infty, \mathsf{odd})} = \varprojlim X^{(2n+1)}$.

…and they have a nice conversation…

In the category theory language, one may define the category of Banach spaces with contractions as morphisms, and denote it by $\mathbf{Ban}_1$. There is a functor $\mathsf{Dir}:\mathbf{Ban}_1 \to \mathbf{Ban}_1$, which assigns to each Banach space its direct limit of even biduals —that is, $\mathsf{Dir}(X) = X^{(\infty, \mathsf{even})}$—, and to each contraction $T:X\to Y$ the induced contraction $\mathsf{Dir}(T):\mathsf{Dir}(X) \to \mathsf{Dir}(Y)$.

• Clearly, $\mathsf{Dir}(X) = X$ if $X$ is a reflexive Banach space.
• In the case of the James space $\mathfrak{J}$ which is non-reflexive but isometrically isomorphic to its bidual, we have that $\mathsf{Dir}(\mathfrak{J}) = \mathfrak{J} \oplus \mathfrak{R}$, where $\mathfrak{R} = \mathbb{R}^{\mathbb{N}}$ is normed by induced norm on $\mathfrak{J} \oplus \operatorname{span}\{e_1, \dots, e_n\}$ for each $n$. ⁶.

“Naturally”, there is an analogous inverse (or projective) limit construction for the odd biduals, which we denote by $\mathsf{Inv}:\mathbf{Ban}_1 \to \mathbf{Ban}_1$, and it has given by the adjoint functor of $\mathsf{Dir}$. Thus, we can set $\mathsf{Inv}(X) = X^{(\infty, \mathsf{odd})}$.

…about spectral properties…

One of the highlights of the paper is the study of the spectral properties of the induced operators $\mathsf{Dir}(T)$ and $\mathsf{Inv}(T)$, for a given contraction $T:X\to X$. In particular, the author shows that the spectrum is preserved in both cases: $\sigma(\mathsf{Dir}(T)) = \sigma(T) = \sigma(\mathsf{Inv}(T)).$

Under approximation properties of the space $X$, one can also show that these limits preserve compactness of operators and even Fredholm index.

…and make plans to meet again.

One of the main hopes of mine—and the reason I write about this topic—is to generalize these constructions to general Locally Convex Topological Vector Spaces (LCTVS)! Albeit this does not seem trivial at all, I think prudent definitions of choice functors for “cherry-picking” topologies on the duals may lead to interesting results, and well-posed definitions of direct and inverse limits in this more general setting.

It is clear, that, under some trivial choices, these limits trivialize due to the characterization of the biduals.

As this is pure speculation, I will leave this as an open question for future research.

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Footnotes

1. Can we consider more general settings? Usually yes. For instance, see the referred books. ↩

2. Bourbaki, Nicolas. Topological vector spaces: Chapters 1–5. Springer Science & Business Media, 2013. ↩

3. Schaefer, Helmut H. “Topological Vector Spaces.” Graduate Texts in Mathematics. Springer New York, NY, 1999. ↩

4. Some cases are worth mentioning: the weak topology, the Mackey topology, and the strong topology. The terminology is not great, and it can be confusing at times… These constructions can be thought in terms of their generating basis and/or seminorms. ↩

5. Gwizdek, Sebastian. “Isometric direct limits of bidual Banach spaces.” arXiv preprint arXiv:2205.14385 (2022). ↩ ↩²

6. Here, I am being lazy to provide the details. ↩