It is proved that every infinite-dimensional Banach space \(X\) of cardinality \(m\) admits both a strictly descending chain and a strictly ascending chain of dense linear subspaces of length \(m\).
We work over \(\Bbb K\in\{\Bbb R,\Bbb C\}\). Fix once and for all a countable dense subfield \(F\subset\Bbb K\), namely \(F=\Bbb Q\) if \(\Bbb K=\Bbb R\) and \(F=\Bbb Q(i)\) if \(\Bbb K=\Bbb C\).
Lemma 1 (Cardinality of vector spaces).If \(V\) is a vector space over a field \(\Bbb F\) of cardinality \(|\Bbb F|=\lambda\), and \(\dim_{\Bbb F}V=\kappa\) is infinite, then \[|V|=\max\{\lambda,\kappa\}.\]
Proof. If \(B\) is a basis of \(V\) of size \(\kappa\), every vector is a finite \(\Bbb F\)-linear combination of elements of \(B\), giving \(|V|\le \bigcup_{n<\omega} (\lambda^n\!\cdot\!\kappa^n)=\max\{\lambda,\kappa\}\). The reverse inequality is immediate since both \(B\) and \(\Bbb F\) embed into \(V\). ◻
Lemma 2(Open sets have full cardinality). If \(X\) is an infinite-dimensional normed space and \(U\subset X\) is a nonempty open set, then \(|U|=|X|\).
Proof. Let \(B=\{x:\|x\|<1\}\). Since \(X=\bigcup_{n<\omega} nB\), \(|X|=|B|\). Every nonempty open set contains a translate of some \(\lambda B\), hence has size \(|B|=|X|\). ◻
Remark 1(Historical note).The existence of discontinuous linear functionals on infinite-dimensional Banach spaces goes back to the early development of functional analysis (see Banach [1]). It follows from the existence of Hamel bases: extend a linear functional from a basis arbitrarily, which need not be continuous. The kernels of such functionals provide natural examples of dense, proper subspaces of Banach spaces.
Lemma 3(Proper dense F-linear subspace).. Let \(X\) be an infinite-dimensional Banach space over \(\Bbb K\) and \(F\subset \Bbb K\) a countable dense subfield. Then there exists a proper dense \(F\)-linear subspace \(L_F\subsetneq X\).
Proof. Fix a base \(\{U_\xi:\xi<\kappa\}\) of nonempty open sets of \(X\), where \(\kappa=\mathrm{dens}(X)\). Choose \(y\in X\setminus\{0\}\). By recursion build \(S=\{x_\xi:\xi<\kappa\}\) with
(1) \(x_\xi\in U_\xi\),
(2) \(x_\xi\notin \operatorname{span}_F(S\cap \xi)\),
(3) \(x_\xi\notin y+\operatorname{span}_F(S\cap \xi)\).
At stage \(\xi\), \(W_\xi:=\operatorname{span}_F(S\cap \xi)\) has size \(\le \max\{\aleph_0,|\xi|\}<|X|\) since \(F\) is countable. By Lemma 2, \(|U_\xi|=|X|\), so \(U_\xi\setminus(W_\xi\cup(y+W_\xi))\neq\varnothing\); choose \(x_\xi\) there. Let \(L_F=\operatorname{span}_F(S)\). Then \(S\) meets every basic open set, so \(L_F\) is dense. Condition (3) guarantees \(y\notin L_F\), hence \(L_F\neq X\). ◻
Lemma 4(Discontinuous functionals with dense kernels). There exists a discontinuous linear functional \(f:X\to\Bbb K\) whose kernel is a proper dense \(\Bbb K\)-linear subspace of \(X\).
Proof. First build a dense \(\Bbb K\)-linear subspace. Let \(\{U_\xi:\xi<\kappa\}\) be a base of nonempty open sets with \(\kappa=\mathrm{dens}(X)\). Construct by transfinite recursion a set \(S=\{s_\xi:\xi<\kappa\}\subset X\) such that:
(i) \(s_\xi\in U_\xi\), and
(ii) \(s_\xi\notin \operatorname{span}_{\Bbb K}(S\cap\xi)\).
At stage \(\xi\), \(\operatorname{span}_{\Bbb K}(S\cap\xi)\) has cardinal \(<|X|\), while \(|U_\xi|=|X|\) by Lemma 2, so such a choice is possible. Then \(L=\operatorname{span}_{\Bbb K}(S)\) is dense and, choosing \(y\notin L\), is proper.
Select a sequence \((y_n)_{n\ge1}\subset X\setminus L\) with \(\|y_n\|\le 2^{-n}\) and whose cosets in \(X/L\) are linearly independent. Extend \(L\cup\{y_n:n\ge1\}\) to a Hamel basis \(B\) of \(X\). Define \(f\) on \(B\) by \(f(y_n)=n\) and \(f\equiv 0\) on \(B\setminus\{y_n:n\ge1\}\), and extend linearly. Then \(y_n\to 0\) while \(f(y_n)=n\to\infty\), so \(f\) is discontinuous. Since \(f\) vanishes on \(L\) and \(L\) is dense, \(\ker f\) is a proper dense subspace. ◻
Theorem 1 (Descending chain).Let \(X\) be an infinite-dimensional Banach space over \(\Bbb K\) with \(|X|=m\). Then there is a strictly descending chain \((D_\alpha)_{\alpha<m}\) of dense subspaces of \(X\).
Proof. Let \(L_0\subsetneq X\) be a proper dense subspace (Lemma 4). Choose a Hamel basis \(B_L\) of \(L_0\) and extend it to a basis \(B\) of \(X\). Pick a subset \(C=\{v_\beta:\beta<m\}\subset B\setminus B_L\). Define \[D_\alpha := L_0 \oplus \operatorname{span}_{\Bbb K}\{v_\beta:\beta\ge\alpha\},\qquad \alpha<m.\]
Each \(D_\alpha\) contains \(L_0\), hence is dense; and \(v_\alpha\in D_\alpha\setminus D_{\alpha+1}\), so \(D_{\alpha+1}\subsetneq D_\alpha\). ◻
Theorem 2 (Ascending chain).Let \(X\) be an infinite-dimensional Banach space over \(\Bbb K\) with \(|X|=m\). Then there is a strictly ascending chain \((E_\alpha)_{\alpha<m}\) of dense subspaces of \(X\).
Proof. Let \(L_0\subsetneq X\) be a proper dense subspace (Lemma 4). Choose a Hamel basis \(B_L\) of \(L_0\) and extend it to a basis \(B\) of \(X\). Pick a subset \(C=\{v_\beta:\beta<m\}\subset B\setminus B_L\). Define \[E_\alpha := L_0 \oplus \operatorname{span}_{\Bbb K}\{v_\beta:\beta<\alpha\},\qquad \alpha<m.\]
Each \(E_\alpha\) contains \(L_0\), hence is dense; and \(v_\alpha\in E_{\alpha+1}\setminus E_\alpha\), so \(E_\alpha\subsetneq E_{\alpha+1}\). ◻
The constructions above suggest several natural extensions and applications:
1. Order-theoretic invariants.
The poset of dense subspaces of \(X\) (ordered by inclusion) admits chains of maximum length \(|X|\). One may ask about antichains, cofinality, and possible order types of maximal chains, in the spirit of Todorčević’s set-theoretic analysis of substructures [2, 3].
2. Density character vs. cardinality.
Our chains have length \(|X|\), the cardinality of the Banach space. It is natural to ask whether chains of length equal to the density character \(\mathrm{dens}(X)\) always exist, and to compare these two invariants systematically. [Note that our results guarantee chains of length \(|X|\), not necessarily of length \(\mathrm{dens}(X)\). If \(\mathrm{dens}(X)\ge \mathfrak{c}\), then indeed \(|X|=\mathrm{dens}(X)\) and our construction already yields chains of length \(\mathrm{dens}(X)\). However, if \(X\) is separable then \(\mathrm{dens}(X)=\aleph_0\) while \(|X|=\mathfrak{c}\), and our theorem produces chains of length \(\mathfrak{c}\). The existence of chains of length \(\aleph_0\) in separable spaces is of course easy (one can enumerate a countable dense set and span partial subsets), but it does not immediately follow from our argument. Thus the precise relationship between chains of length \(|X|\) and those of length \(\mathrm{dens}(X)\) deserves separate attention.]
3. Lineability and spaceability.
Dense subspaces often appear in lineability results (e.g. Aron–Gurariy–Seoane-Sepúlveda [4] and the monograph [5]). Our chains may refine such results by distinguishing many degrees of density simultaneously.
4. Duality.
Dense subspaces have trivial annihilators in the dual: if \(E\subset X\) is dense then \(E^\perp=\{0\}\subset X^*\). Understanding the induced chains of annihilators may illuminate structure in dual Banach spaces and relates to broader themes on long chains of subspaces studied via combinatorial set theory [2, 3].
5. Descriptive set theory.
In separable Banach spaces, the collection of dense subspaces is a coanalytic subset of the Effros Borel space of closed subspaces (see Kechris [6]). Our maximal chains suggest this set has high order-theoretic complexity. Connections with classical separation results and structural analysis (Louveau [7]; Rosendal [8, 9]) may be fruitful.
6. Beyond Banach spaces.
The arguments apply almost verbatim to normed spaces and more generally to locally convex topological vector spaces. One may investigate whether similar chains exist in non-locally convex settings or under weaker topologies.
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Bernal-González, L., Pellegrino, D., & Seoane-Sepúlveda, J. B. (2016). Lineability: The Search for Linearity in Mathematics. CRC Press.
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Rosendal, C. (2008). Infinite asymptotic games. Annals of Pure and Applied Logic, 155(3), 184–192.
Rosendal, C. (2022). Coarse Geometry of Topological Groups (Vol. 223, Cambridge Tracts in Mathematics). Cambridge University Press.
