Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\aleph_0}$ has a Hamel basis of cardinality continuum (OK, I do know it cannot be smaller for an inf.-dim. Banach space)?

It was proved by G.W. Mackey, in On infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57 (1945), 155-207, see Theorem I-1, p.158, that an infinite-dimensional Banach space has Hamel dimension at least $\mathfrak{c} = 2^{\aleph_0}$. A short proof can also be found in H. Elton Lacey, The Hamel Dimension of any Infinite Dimensional Separable Banach Space is $c$, Amer. Math. Mon. 80 (1973), 298.

Moreover, a vector space over $\mathbb{R}$ of cardinality $\kappa \gt \mathfrak{c}$ has dimension $\kappa$ by a theorem of Löwig, Über die Dimension linearer Räume, Studia Math. 5 (1934), pp. 18–23.

Added: By combining these two facts we get the crisp statement (as given by Halbeisen and Hungerbühler in the paper Jonas linked to in a comment): “The Hamel dimension of an infinite-dimensional Banach space is equal to its cardinality.”

Finally, $\ell^1(\mathbb{R})$ embeds isometrically into $\ell^\infty(\mathbb{N})$, so its dimension is at most the cardinality of $\ell^\infty(\mathbb{N})$ which is $\mathfrak{c} = \#(\mathbb{R}^{\aleph_0})$.

To answer your question whether a Banach space $X$ of density $\mathfrak{c} = 2^{\aleph_0}$ must have dimension $\mathfrak{c}$ and whether this is a consequence of knowing the dimension of $\ell^1(\mathbb{R})$: yes.

This is because it suffices to pick a dense subset $S$ of cardinality $\mathfrak{c}$ in the unit sphere of $X$, then choose a bijection $\mathbb{R} \to S$ and send the standard basis $(e_t)_{t \in \mathbb{R}}$ of $\ell^1(\mathbb{R})$ to $S$. This map extends to a map $\ell^1(\mathbb{R}) \to X$ which is onto by the Banach–Schauder theorem (usually proved as part of the open mapping theorem: if a continuous linear map sends the unit ball of $Y$ densely into the unit ball of $X$ then it is onto).

• Life in a universe where inf. dim. Banach spaces have a Hamel basis?? Just crazy... What's next? Ultrafilters over $\mathbb N$ and choice function on socks?? – Asaf Karagila May 5 '12 at 22:25
• :) ${}{}{}{}{}{}$ – t.b. May 5 '12 at 22:26

Theorem. Let $X$ be an infinite dimensional Banach space. Then $\,\mathrm{dim}\,X\ge 2^{\aleph_0}$.

Sketch of Proof. Based on M.G. McKay's proof. Let $\{w_n : n\in\mathbb N\}\subset X$ be a linearly independent set.

Step A. Using Hahn-Banach, we shall construct another linearly independent set $\{v_n : n\in\mathbb N\}\subset X$, and a set of linear functionals $\{v^*_n:n\in\mathbb N\}\subset X^*$, such that $$\mathrm{span}\{v_1,\ldots,v_n\}=\mathrm{span}\{w_1,\ldots,w_n\}, \quad \text{for all n\in\mathbb N,}$$ $\|v_i^*\|=1$, for all $i\in\mathbb N$, and $$v_i^*(v_j)=\delta_{ij}, \quad \text{for all i,j\in \mathbb N.}$$ This is done inductively. Define $v_1=w_1/\|w_1\|$, and $v_1^*(v_1)=1$, and extend, using Hahn-Banach to $X$, so that $\|v_1^*\|=1$. Assume that $v_1,\ldots,v_k$ and $v_1^*,\ldots,v_1^*$, have been defined so that $$\mathrm{span}\{v_1,\ldots,v_k\}=\mathrm{span}\{w_1,\ldots,w_k\},\quad \|v_i^*\|=1\,\,\text{and}\,\,\,v_i^*(v_j)=\delta_{ij}, \quad \text{for all \,i,j=1,\ldots,k.}$$ Then let $v_{k+1}=w_{k+1}-\sum_{j=1}^k v_j^*(w_{k+1})v_j\in \bigcap_{j=1}^k\mathrm{ker}\,v_j^*$, and define the functional $v_{k+1}^*$, so that $v_{k+1}^*(v_j)=\delta_{k+1,j}$, for $j=1,\ldots,k+1$, and extend it via Hahn-Banach to $X$, and to keep its norm unit we suitably rescale $v_{k+1}$.

Step B. Let $\mathcal S\subset\mathbb P(\mathbb N)$, such that that: If $A,B\in\mathcal S$, and $A\ne B$, then $\,\rvert A\cap B\rvert<\aleph_0$, while $\lvert\mathcal S\rvert=2^{\aleph_0}$. Then define $$v_A=\sum_{n\in A}2^{-n}v_n, \quad A\in\mathcal S.$$ It is readily shown that the set $\{v_A:A\in\mathcal S\}\subset X$ is linearly independent and equi-numerous to $\mathbb R$.

• I think that it is worth mentioning that this is basically the proof from H. Elton Lacey's paper mentioned in the other answer: jstor.org/stable/2318458 – Martin Sleziak Nov 25 '15 at 7:09
• @MartinSleziak: This is true! – Yiorgos S. Smyrlis Nov 25 '15 at 7:19