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Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer. If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a sequence $\{\alpha_n\}_{n=1}^\infty$ of strictly positive real scalars such that $\lim_{n\to\infty} \alpha_n x_n = 0$. In the aforementioned answer, I […]

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Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer. If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a sequence $\{\alpha_n\}_{n=1}^\infty$ of strictly positive real scalars such that $\lim_{n\to\infty} \alpha_n x_n = 0$. In the aforementioned answer, I […]

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Ask Mathematics

Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer. If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a sequence $\{\alpha_n\}_{n=1}^\infty$ of strictly positive real scalars such that $\lim_{n\to\infty} \alpha_n x_n = 0$. In the aforementioned answer, I […]