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edit approved Jan 13, 2016 at 18:25
Zbigniew
edit approved Jan 13, 2016 at 18:25
Zbigniew
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This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^\*$$H^{\star}$, we have the inclusions $$X^\* \hookrightarrow H^\* \cong H \hookrightarrow X.$$$$X^{\star} \hookrightarrow H^{\star} \cong H \hookrightarrow X.$$

My question: Is $i^\* : X^\* \hookrightarrow H^\*$$i^{\star} : X^{\star} \hookrightarrow H^{\star}$ a dense injection?

This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^\*$, we have the inclusions $$X^\* \hookrightarrow H^\* \cong H \hookrightarrow X.$$

My question: Is $i^\* : X^\* \hookrightarrow H^\*$ a dense injection?

This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^{\star}$, we have the inclusions $$X^{\star} \hookrightarrow H^{\star} \cong H \hookrightarrow X.$$

My question: Is $i^{\star} : X^{\star} \hookrightarrow H^{\star}$ a dense injection?

Added "separable"
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Tom LaGatta
Tom LaGatta
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This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, let $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^\*$, we have the inclusions $$X^\* \hookrightarrow H^\* \cong H \hookrightarrow X.$$

My question: Is $i^\* : X^\* \hookrightarrow H^\*$ a dense injection?

This seems like a really simple question, but I'm struggling with it. Let $X$ be a Banach space, let $H$ be a Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^\*$, we have the inclusions $$X^\* \hookrightarrow H^\* \cong H \hookrightarrow X.$$

My question: Is $i^\* : X^\* \hookrightarrow H^\*$ a dense injection?

This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^\*$, we have the inclusions $$X^\* \hookrightarrow H^\* \cong H \hookrightarrow X.$$

My question: Is $i^\* : X^\* \hookrightarrow H^\*$ a dense injection?

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Tom LaGatta
Tom LaGatta
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Dense inclusions of Banach spaces and their duals

This seems like a really simple question, but I'm struggling with it. Let $X$ be a Banach space, let $H$ be a Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, continuous embedding of $H$ into $X$. (This is the abstract Wiener space construction due to Gross, hence the [pr.probability] tag) If we associate $H$ with its dual $H^\*$, we have the inclusions $$X^\* \hookrightarrow H^\* \cong H \hookrightarrow X.$$

My question: Is $i^\* : X^\* \hookrightarrow H^\*$ a dense injection?