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Given an arbitrary infinite dimensional Banach space $X$, can we deduce that it's dimension $\dim(X)$ (the cardinality of one of its Hamel bases) is less or equal of the dimension $\dim(X^{\ast})$ of its dual space (the space of all continuous linear functionals $f:X\to\mathbb{R}$)?

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This is an interesting question which has been unaddressed for a long time, so I'll give it a shot.

Lets denote the cardinality of a set $A$ by $|A|$. For a normed space $X$ we define its density character $d(X)$ as the smallest cardinality of its dense sets, that is $d(X)=\min\{|D|: D\subseteq X, \overline{D}=X\}$. In particular a separable normed space $X$ has $d(X)=\aleph_0$.

We need the following three lemmas:

Lemma 1: If $X$ is an infinite dimensional vector space over $\mathbb{R}$ and $\dim X\geq |\mathbb{R}|$, then $\dim X=|X|$.

A proof of Lemma 1 can be found here.

Lemma 2: Let $X$, $Y$ be infinite dimensional Banach spaces with $d(X)\leq d(Y)$. Then $|X|\leq |Y|.$

You can find a proof here.

For the last step, it is known from functional analysis that when $X^*$ is separable, then so is $X$. If you check the proof carefully you'll realise that what is actually proven is the following:

Lemma 3: Let $X$ be a normed space. Then $d(X)\leq d(X^*)$.

Proof. Let $\{x_a^*: a\in A\}$ be a dense subset of $S_{X^*}$ of cardinality $|A|=d(X^*)$. For every $a\in A$ we pick an $x_a\in B_X$ with $x_a^*(x_a)>\tfrac{1}{2}$ and set $Y= \langle \{x_a: a\in A\} \rangle$. The set $Y$ is dense in $X$: Otherwise, there would exist an $f \in X^*$ with $\|f\|=1$ such that $f(y)=0$ for every $y\in Y$. But then for every $a\in A$, \begin{eqnarray*} \|f-x_a^*\|\geq |f(x_a)-x_a^*(x_a)|=\frac{1}{2}, \end{eqnarray*} which implies that $f\notin \overline{\{x_a^*: a\in A\}}=S_{X^*}$, a contradiction. So $X$ contains a dense subset of cardinality $|Y|\leq |A|$, therefore $d(X)\leq |A|=d(X^*)$.

Combining the previous lemmas, we get an affirmative answer to remilt's question.

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    $\begingroup$ Nice! Thank you tree detective, it's been two years since I have made this question and had completely forgot it since it never got much attention! $\endgroup$ – remilt Nov 6 '16 at 19:55
  • $\begingroup$ Just to confirm, does this argument use Hahn-Banach? $\endgroup$ – MichaelGaudreau Sep 12 '18 at 1:52
  • $\begingroup$ @MichaelGaudreau Yes, the existence of an $f$ with norm one that vanishes on $\overline{Y}$ in the proof of Lemma 3, is a consequence of the Hahn Banach theorem. $\endgroup$ – tree detective Sep 12 '18 at 10:01
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That is a great answer to a very interesting problem. Given your current solution, I would like to suggest a possible generalization to the case of normed spaces.

Since in the finite dimensional case everything is working well, let $X$ be an infinite dimensional normed space and consider the canonical embedding of $X$ to its double dual $X^{**}$ given by:

$T:X\rightarrow X^{**},\hspace{10pt} T(x)=T_x \hspace{5pt}$ where for $x\in X,$ we define $T_x:X^*\rightarrow\mathbb{R}$ such that $ \hspace{7pt} {T_x}(x^*)=x^*(x)$

Of course, T is an isometric embedding.

Consider the spaces:

$T(X)\hspace{5pt}$ (which is isometric to $X$ and obviously dense in $\overline{T(X)}$) and

$\overline{T(X)}\hspace{5pt}$ (which is a Banach space, as it is a closed subspace of the Banach space $X^{**}$)

(This process is of course standard when considering the completion of a normed space $X$).

Since $\overline{T(X)}$ is a Banach space, by your argument we must have that $\dim(\overline{T(X)})\leq \dim(({\overline{T(X)}})^*)$

But, since $T(X)$ is dense in $\overline{T(X)},$ we must have that the spaces ${(T(X))}^*$ and $({\overline{T(X)}})^*$ are isometric.

(Indeed, consider this result when stated in the following more general fashion:

Let $X$ be a normed space, $Z$ a dense subspace of $X$ and $Y$ a Banach space. Then, the spaces $\mathcal{B}(X,Y)$ and $\mathcal{B}(Z,Y)$ are isometric.

I will be glad to give hints to the proof of this fact, if anyone is interested).

Now combine all the previous arguments together to get:

$\dim(X)= \dim(T(X))\leq \dim({\overline{T(X)}})\leq \dim(({\overline{T(X)}})^*) = \dim(({T(X)})^*)= \dim(X^*)$

(Notice that we have also implicitly used the fact that if two spaces are isometric, then their duals must be isometric as well).

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    $\begingroup$ Wow! Suddenly a lot of motion around this old question. Now I wonder if we have a space with dimension strictly smaller than its dual, and we consider its orbit when * acts on it. Does there exists such a space, for which the dimension finally becomes constant? If yes when we end up with a reflexive space or even with a self dual space? $\endgroup$ – remilt Nov 8 '16 at 22:26
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    $\begingroup$ If you could end up with a reflexive space, this would imply that your original space would be reflexive as well (and actually the whole sequence $X, X^*, X^{**}\ldots$), since a Banach space is reflexive if and only if its dual is reflexive. This would contradict the fact that $\dim X< \dim X^*$. But the question is still valid, I mean whether the dimension can become constant. The only example of a space $X$ with $\dim X <\dim X^*$ that I know of, is $X=\ell_\infty$, but the dimension of the sequence of the duals is strictly increasing. $\endgroup$ – tree detective Nov 9 '16 at 19:07
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    $\begingroup$ Sorry I guess I have ages to even touch Functional analysis. Can't we have $\dim X<\dim X^*=\dim X^{**}=\dim X^{***}=\dim X^{****}$ with $X^*=X^{***}$, and $X^{**}=X^{****}$? I mean we may have $X$ not even a pre-dual no? Also do you have a reference about the fact that the sequence of the duals of $l_{\infty}$ is strictly increasing? $\endgroup$ – remilt Nov 10 '16 at 22:11
  • $\begingroup$ All I am saying is that none of these spaces can be reflexive, provided that $X$ is a Banach space. But they may end up to be isomorphic, or even isometrically isomorphic, I can't rule out this possibility. I guess that by equality you mean the latter. If $X$ is not reflexive you can end up with a reflexive space of course (for example, $X=(c_{00}, \|\cdot\|_p)$. By the way, reading your first comment, I started to wonder whether a self dual space is reflexive. Is this true? I haven't managed to show it. $\endgroup$ – tree detective Nov 13 '16 at 20:33
  • $\begingroup$ About the duals of $\ell_\infty$ I wasn't exactly correct. To be more precise the sequence $\ell_\infty, \ell_\infty^{**}, \ell_\infty^{****}, \ldots,$ has strictly increasing dimensions by Theorem 2.1 of Rosenthal's paper: carma.newcastle.edu.au/jon/Preprints/Books/CUP/Material/… If $\Delta $ is a set, then $\ell_\infty(\Delta)^{**} \simeq \ell_\infty(\Gamma)$, where $\Gamma$ is a set of cardinality $|\Gamma|=2^{\dim \ell_\infty(\Delta)}$. The same theorem also describes $\ell_\infty(\Delta)^*$, but I can't say for sure whether its dimension lies strictly in between. $\endgroup$ – tree detective Nov 13 '16 at 20:45

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