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I have recently noticed$^1$ that in models of ZF+DC+"All sets of real numbers have the Baire property" (e.g. Solovay's model or Shelah's model mentioned by Andreas) there is a very interesting property for Banach spaces:
If $V$ is a Banach space, $W$ is a normed space and $T\colon V\to W$ is linear then $T$ is continuous.
In particular this means that if $V$ is a Banach space over $\mathbb R$ then every functional is continuous. This means that the algebraic dual of a Banach space is the same as the continuous dual of the space.
For example, $\ell_p$ for $p\in(1,\infty)$ have this property, using DC we can develop the basic tools of functional analysis just fine (except Hahn-Banach, though). This implies that $\ell_p$ is reflexive in the algebraic sense, not only in the topological sense.
We further remark$^2$ that the assertion $\ell_1^\prime\subsetneq\ell^\infty$ \ell_1\subsetneq\ell_\infty^\prime$(where the$\prime$denotes all the continuous functionals, but in our model these are all the functionals) implies the negation of "All sets of real numbers have the Baire property". The result is that$\ell_1$is also reflexive in the algebraic sense and in the topological sense. So with this in mind we have$V_p=\ell_p\oplus\ell_q$(where$\frac1p+\frac1q=1$) is a self-dual as well algebraically reflexive space, and for distinct$p_1,p_2\in[1,2]$we have$V_{p_1}\ncong V_{p_2}$as well, so there are many non-isomorphic examples for this. Notes: 1. After sitting to write all the details of the above I found out it is mentioned in Eric Schechter's book, Handbook of Analysis and its Foundations in the last section of chapter 27. 2. The above reference does not discuss the case of$p=1$, which is discussed later in the end of chapter 29. 1 I have recently noticed$^1$that in models of ZF+DC+"All sets of real numbers have the Baire property" (e.g. Solovay's model or Shelah's model mentioned by Andreas) there is a very interesting property for Banach spaces: If$V$is a Banach space,$W$is a normed space and$T\colon V\to W$is linear then$T$is continuous. In particular this means that if$V$is a Banach space over$\mathbb R$then every functional is continuous. This means that the algebraic dual of a Banach space is the same as the continuous dual of the space. For example,$\ell_p$for$p\in(1,\infty)$have this property, using DC we can develop the basic tools of functional analysis just fine (except Hahn-Banach, though). This implies that$\ell_p$is reflexive in the algebraic sense, not only in the topological sense. We further remark$^2$that the assertion$\ell_1^\prime\subsetneq\ell^\infty$(where the$\prime$denotes all the continuous functionals, but in our model these are all the functionals) implies the negation of "All sets of real numbers have the Baire property". The result is that$\ell_1$is also reflexive in the algebraic sense and in the topological sense. So with this in mind we have$V_p=\ell_p\oplus\ell_q$(where$\frac1p+\frac1q=1$) is a self-dual as well algebraically reflexive space, and for distinct$p_1,p_2\in[1,2]$we have$V_{p_1}\ncong V_{p_2}$as well, so there are many non-isomorphic examples for this. Notes: 1. After sitting to write all the details of the above I found out it is mentioned in Eric Schechter's book, Handbook of Analysis and its Foundations in the last section of chapter 27. 2. The above reference does not discuss the case of$p=1\$, which is discussed later in the end of chapter 29.