What is an example of a Banach space that is not a BK-space?
A normed sequence space $X$ (with projections $p_n$) is a BK Space if $X$ is Banach space and for all natural numbers $n$, $p_n(\bar{x}) = x_n$ is continuous, for all $\bar{x}$ in $X$.
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$\begingroup$ You haven't defined $p_n$. $\endgroup$– YCorCommented May 9 at 7:18
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$\begingroup$ @YCor I believe they are the coordinate projections, since a BK-space is a Banach space whose elements are sequences (plus the continuity condition) $\endgroup$– David Roberts ♦Commented May 9 at 7:21
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1$\begingroup$ @Maulana Do you want a Banach space that is also a sequence space? Otherwise any Banach space that is not embeddable in a sequence space might count (warning: I'm not a functional analyst!) $\endgroup$– David Roberts ♦Commented May 9 at 7:23
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It seems that a BK-space $X$ should have a continuous linear inclusion (in particular, injective map) into $\mathbb C^{\mathbb N}$ (with the product topology making it a Fréchet space). This is certainly impossible if the cardinality (or Hamel-dimension) of $X$ is bigger than that of $\mathbb C^{\mathbb N}$. A concrete example is the Banach space $X$ of all bounded functions from $\mathbb R\to\mathbb C$.
answered May 9 at 10:04
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$\begingroup$ Yes, it is elementary that a Fréchet space $X$ is a BK space if and only if there is a continuous linear injection from $X$ into $\mathbb C^{\mathbb N}$ (with the product topology) if and only if $X^*$ is weak$^*$ separable. $\endgroup$ Commented May 11 at 18:22