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Martin Sleziak
Martin Sleziak
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The answer to this question is YES -- but it is useless!

In fact, a theorem of Valdivia (which you can find, e.g., as Theorem 6.5.8Theorem 6.5.8 in the book Barrelled Locally Convex Spaces of Bonet and Perez-Carreras) states that given any infinite-dimensional separable Banach space $F$, one can represent any ultrabornological locally convex space $E$ (this means, that $E$ has some representation as an inductive limit of Banach spaces) which does not carry the finest locally convex topology as an inductive limit of Banach spaces isomorphic to $F$ such that, in addition,the linking maps beweenbetween the steps are nuclear operators.

This theorem is rather useless because it is too good: Every ultrabornological space has such a representation and thus you cannot deduce any nice properties from it.

The answer to this question is YES -- but it is useless!

In fact, a theorem of Valdivia (which you can find, e.g., as Theorem 6.5.8 in the book Barrelled Locally Convex Spaces of Bonet and Perez-Carreras) states that given any infinite-dimensional separable Banach space $F$, one can represent any ultrabornological locally convex space $E$ (this means, that $E$ has some representation as an inductive limit of Banach spaces) which does not carry the finest locally convex topology as an inductive limit of Banach spaces isomorphic to $F$ such that, in addition,the linking maps beween the steps are nuclear operators.

This theorem is rather useless because it is too good: Every ultrabornological space has such a representation and thus you cannot deduce any nice properties from it.

The answer to this question is YES -- but it is useless!

In fact, a theorem of Valdivia (which you can find, e.g., as Theorem 6.5.8 in the book Barrelled Locally Convex Spaces of Bonet and Perez-Carreras) states that given any infinite-dimensional separable Banach space $F$, one can represent any ultrabornological locally convex space $E$ (this means, that $E$ has some representation as an inductive limit of Banach spaces) which does not carry the finest locally convex topology as an inductive limit of Banach spaces isomorphic to $F$ such that, in addition,the linking maps between the steps are nuclear operators.

This theorem is rather useless because it is too good: Every ultrabornological space has such a representation and thus you cannot deduce any nice properties from it.

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Jochen Wengenroth
Jochen Wengenroth
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The answer to this question is YES -- but it is useless!

In fact, a theorem of Valdivia (which you can find, e.g., as Theorem 6.5.8 in the book Barrelled Locally Convex Spaces of Bonet and Perez-Carreras) states that given any infinite-dimensional separable Banach space $F$, one can represent any ultrabornological locally convex space $E$ (this means, that $E$ has some representation as an inductive limit of Banach spaces) which does not carry the finest locally convex topology as an inductive limit of Banach spaces isomorphic to $F$ such that, in addition,the linking maps beween the steps are nuclear operators.

This theorem is rather useless because it is too good: Every ultrabornological space has such a representation and thus you cannot deduce any nice properties from it.