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YCor
YCor
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Breaking Up Dense Setup dense subset in Nonnon-Separable Spaceseparable space

Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent setsubset. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X_i\}_{i \in I}$ of $X$ with the property that:

  • $D\cap X_i$ is dense in $X_i$,
  • $\bigcup_{i \in I} X_i$ is dense in $X$?

Breaking Up Dense Set in Non-Separable Space

Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent set. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X_i\}_{i \in I}$ of $X$ with the property that:

  • $D\cap X_i$ is dense in $X_i$,
  • $\bigcup_{i \in I} X_i$ is dense in $X$?

Breaking up dense subset in non-separable space

Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X_i\}_{i \in I}$ of $X$ with the property that:

  • $D\cap X_i$ is dense in $X_i$,
  • $\bigcup_{i \in I} X_i$ is dense in $X$?
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ABIM
ABIM
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Breaking Up Dense Set in Non-Separable Space

Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent set. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X_i\}_{i \in I}$ of $X$ with the property that:

  • $D\cap X_i$ is dense in $X_i$,
  • $\bigcup_{i \in I} X_i$ is dense in $X$?