Let $E$ be an infinite dimensional Banach space, let $E^{\*}$ denote its continuous (i.e., Banach space) dual, and let $E^'$ be its algebraic dual. Clearly, $E^{\*}$ is a proper vector subspace of $E'$. Now, let us suppose that $E^{\*}$ and $E'$ are algebraically isomorphic (i.e., as vector spaces). Does it follow that $E$ contains an isomorph of the Banach space $\ell_{1}(\mathbb{R})$ ?

[By "*isomorph of X" I mean a closed linear subspace both algebraically and topologically isomorphic to X.]

P.S. This is under ZFC + CH.

P.P.S. The answer is affirmative if $E$ is the dual of a separable [infinite-dimensional] Banach space. It would be interesting to see if it is also affirmative when $E$ is a "nice" space. For instant, a Banach lattice.

Let $E$ be an infinite dimensional Banach space, let $E^{\ast}$ denote its continuous (i.e., Banach space) dual, and let $E'$ be its algebraic dual. Clearly, $E^{\ast}$ is a proper vector subspace of $E'$. Now, let us suppose that $E^{\ast}$ and $E'$ are algebraically isomorphic (i.e., as vector spaces). Does it follow that $E$ contains an isomorph of the Banach space $\ell_{1}(\mathbb{R})$ ?

[By "*isomorph of X" I mean a closed linear subspace both algebraically and topologically isomorphic to X.]

P.S. This is under ZFC + CH.

P.P.S. The answer is affirmative if $E$ is the dual of a separable [infinite-dimensional] Banach space. It would be interesting to see if it is also affirmative when $E$ is a "nice" space. For instant, a Banach lattice.

Let $E$ be an infinite dimensional Banach space, let $E^{\*}$ denote its continuous (i.e., Banach space) dual, and let $E^'$ be its algebraic dual. Clearly, $E^{\*}$ is a proper vector subspace of $E'$. Now, let us suppose that $E^{\*}$ and $E'$ are algebraically isomorphic (i.e., as vector spaces). Does it follow that $E$ contains an isomorph of the Banach space $\ell_{1}(\mathbb{R})$ ?

[By "*isomorph of X" I mean a closed linear subspace both algebraically and topologically isomorphic to X.]

P.S. This is under ZFC + CH.

Let $E$ be an infinite dimensional Banach space, let $E^{\*}$ denote its continuous (i.e., Banach space) dual, and let $E^'$ be its algebraic dual. Clearly, $E^{\*}$ is a proper vector subspace of $E'$. Now, let us suppose that $E^{\*}$ and $E'$ are algebraically isomorphic (i.e., as vector spaces). Does it follow that $E$ contains an isomorph of the Banach space $\ell_{1}(\mathbb{R})$ ?

[By "*isomorph of X" I mean a closed linear subspace both algebraically and topologically isomorphic to X.]

P.S. This is under ZFC + CH.

P.P.S. The answer is affirmative if $E$ is the dual of a separable [infinite-dimensional] Banach space. It would be interesting to see if it is also affirmative when $E$ is a "nice" space. For instant, a Banach lattice.