Return to Question

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same time, A.P.Robertson and W.Robertson in chapter VII of their Topological Vector Spaces write that

if $E$ and $F$ are metrizable locally convex spaces, then each compact set $K\subseteq E\widehat{\otimes} F$ is contained in the closed absolutely convex hull of a sequence $x_n\otimes y_n$, where $x_n\to 0$ and $y_n\to 0$.

Together this implies, that

for each compact set $K\subseteq L_1(\mu\times\nu)$ there are compact sets $S\subseteq L_1(\mu)$ and $T\subseteq L_1(\nu)$ such that $$ K\subseteq S\widehat{\otimes} T $$ (where $S\widehat{\otimes} T$ means closed absolutely convex hull of the set $\{x\otimes y;\ x\in S, \ y\in T\}$ in $L_1(\mu\times\nu)$ ).

I think, that this result can be proved directly, inside the theory of the Banach space $L_1(\mu)$ (and without references to the theory of topological vector spaces, in particular, to Robertsons' proposition). Is that true?

I am asking because I am trying to study the properties of the spaces of universally integrable functions, where I suspect the same must be true (but with a much more complicated topology, which is not metrizable, and therefore one can't apply Robertsons' lemma).

P.S. I consider the case where $\mu$ and $\nu$ are (positive) Radon measures on compact topological spaces.

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same time, A.P.Robertson and W.Robertson in chapter VII of their Topological Vector Spaces write that

if $E$ and $F$ are metrizable locally convex spaces, then each compact set $K\subseteq E\widehat{\otimes} F$ is contained in the closed absolutely convex hull of a sequence $x_n\otimes y_n$, where $x_n\to 0$ and $y_n\to 0$.

Together this implies, that

for each compact set $K\subseteq L_1(\mu\times\nu)$ there are compact sets $S\subseteq L_1(\mu)$ and $T\subseteq L_1(\nu)$ such that $$ K\subseteq S\widehat{\otimes} T, $$ where $S\widehat{\otimes} T$ means closed absolutely convex hull of the set $\{x\otimes y;\ x\in S, \ y\in T\}$ in $L_1(\mu\times\nu)$.

I think, that this result can be proved directly, inside the theory of the Banach space $L_1(\mu)$ (and without references to the theory of topological vector spaces, in particular, to Robertsons' proposition). Is that true?

I am asking because I am trying to study the properties of the spaces of universally integrable functions, where I suspect the same must be true (but with a much more complicated topology, which is not metrizable, and therefore one can't apply Robertsons' lemma).

P.S. I consider the case where $\mu$ and $\nu$ are (positive) Radon measures on compact topological spaces.

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same time, A.P.Robertson and W.Robertson in chapter VII of their Topological Vector Spaces write that

if $E$ and $F$ are metrizable locally convex spaces, then each compact set $K\subseteq E\widehat{\otimes} F$ is contained in the closed absolutely convex hull of a sequence $x_n\otimes y_n$, where $x_n\to 0$ and $y_n\to 0$.

Together this implies, that for each compact set $K\subseteq L_1(\mu\times\nu)$ there are compact sets $S\subseteq L_1(\mu)$ and $T\subseteq L_1(\nu)$ such that $$ K\subseteq S\widehat{\otimes} T $$ (where $S\widehat{\otimes} T$ means closed absolutely convex hull of the set $\{x\otimes y;\ x\in S, \ y\in T\}$ in $L_1(\mu\times\nu)$ ).

I think, that this result can be proved directly, inside the theory of the Banach space $L_1(\mu)$ (and without references to the theory of topological vector spaces, in particular, to Robertsons' proposition). Is that true?

I am asking because I am trying to study the properties of the spaces of universally integrable functions, where I suspect the same must be true (but with a much more complicated topology, which is not metrizable, and therefore one can't apply Robertsons' lemma).

P.S. I consider the case where $\mu$ and $\nu$ are (positive) Radon measures on compact topological spaces.

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same time, A.P.Robertson and W.Robertson in chapter VII of their Topological Vector Spaces write that

if $E$ and $F$ are metrizable locally convex spaces, then each compact set $K\subseteq E\widehat{\otimes} F$ is contained in the closed absolutely convex hull of a sequence $x_n\otimes y_n$, where $x_n\to 0$ and $y_n\to 0$.

Together this implies, that

for each compact set $K\subseteq L_1(\mu\times\nu)$ there are compact sets $S\subseteq L_1(\mu)$ and $T\subseteq L_1(\nu)$ such that $$ K\subseteq S\widehat{\otimes} T $$ (where $S\widehat{\otimes} T$ means closed absolutely convex hull of the set $\{x\otimes y;\ x\in S, \ y\in T\}$ in $L_1(\mu\times\nu)$ ).

I think, that this result can be proved directly, inside the theory of the Banach space $L_1(\mu)$ (and without references to the theory of topological vector spaces, in particular, to Robertsons' proposition). Is that true?

I am asking because I am trying to study the properties of the spaces of universally integrable functions, where I suspect the same must be true (but with a much more complicated topology, which is not metrizable, and therefore one can't apply Robertsons' lemma).

P.S. I consider the case where $\mu$ and $\nu$ are (positive) Radon measures on compact topological spaces.

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same time, A.P.Robertson and W.Robertson in chapter VII of their Topological Vector Spaces write that

if $E$ and $F$ are metrizable locally convex spaces, then each compact set $K\subseteq E\widehat{\otimes} F$ is contained in the closed absolutely convex hull of a sequence $x_n\otimes y_n$, where $x_n\to 0$ and $y_n\to 0$.

Together this implies, that for each compact set $K\subseteq L_1(\mu\times\nu)$ there are compact sets $S\subseteq L_1(\mu)$ and $T\subseteq L_1(\nu)$ such that $$ K\subseteq S\widehat{\otimes} T $$ (where $S\widehat{\otimes} T$ means closed absolutely convex hull of the set $\{x\otimes y;\ x\in S, \ y\in T\}$ in $L_1(\mu\times\nu)$ ).

I think, that this result can be proved directly, inside the theory of the Banach space $L_1(\mu)$ (and without references to the theory of topological vector spaces, in particular, to Robertsons' proposition). Is that true?

I am asking because I am trying to study the properties of the spaces of universally integrable functions, where I suspect the same must be true (but with a much more complicated topology, which is not metrizable, and therefore one can't apply Robertsons' lemma).

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same time, A.P.Robertson and W.Robertson in chapter VII of their Topological Vector Spaces write that

if $E$ and $F$ are metrizable locally convex spaces, then each compact set $K\subseteq E\widehat{\otimes} F$ is contained in the closed absolutely convex hull of a sequence $x_n\otimes y_n$, where $x_n\to 0$ and $y_n\to 0$.

Together this implies, that for each compact set $K\subseteq L_1(\mu\times\nu)$ there are compact sets $S\subseteq L_1(\mu)$ and $T\subseteq L_1(\nu)$ such that $$ K\subseteq S\widehat{\otimes} T $$ (where $S\widehat{\otimes} T$ means closed absolutely convex hull of the set $\{x\otimes y;\ x\in S, \ y\in T\}$ in $L_1(\mu\times\nu)$ ).

I think, that this result can be proved directly, inside the theory of the Banach space $L_1(\mu)$ (and without references to the theory of topological vector spaces, in particular, to Robertsons' proposition). Is that true?

I am asking because I am trying to study the properties of the spaces of universally integrable functions, where I suspect the same must be true (but with a much more complicated topology, which is not metrizable, and therefore one can't apply Robertsons' lemma).

P.S. I consider the case where $\mu$ and $\nu$ are (positive) Radon measures on compact topological spaces.