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ABIM
ABIM
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Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$$$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} \min\left\{d_X(f(y),g(y)),1\right\}, $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fréchet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fréchet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} \min\left\{d_X(f(y),g(y)),1\right\}, $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fréchet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.

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YCor
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Reference Request: Metrizability of Compact-Convergence Topologytopology of compact convergence

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fr'{e}chetFréchet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.

Reference Request: Metrizability of Compact-Convergence Topology

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fr'{e}chet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.

Metrizability of topology of compact convergence

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fréchet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.

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ABIM
ABIM
  • 5.4k
  • 3
  • 19
  • 41

Reference Request: Metrizability of Compact-Convergence Topology

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g):= \sum_{n =0}^{\infty} \frac1{2^n} \sup_{y \in K_n} d_X(f(y),g(y)), $$ where $\{K_n\}_{n \in \mathbb{N}}$ is a countable compact cover of $Y$ and $d_X$ is the metric on $X$. Moreover, if $X$ is Banach then $C(Y,X)$ is a Fr'{e}chet space.

This is easy to show, but I'm looking for a reference to this result. Thanks in advance.