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Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this questionthis question).

What happen to this statement when we are not in $\mathbb{R}^n$ any more, but in an infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question).

What happen to this statement when we are not in $\mathbb{R}^n$ any more, but in an infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question).

What happen to this statement when we are not in $\mathbb{R}^n$ any more, but in an infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

fixed grammar
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Loick
Loick
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Let $K$ be a closed convex cone in $\mathbb{R}^n$. It'sIts dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question).

My question is whatWhat happen to this statement when we are not in $\mathbb{R}^n$ any more, but in aan infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Let $K$ be a closed convex cone in $\mathbb{R}^n$. It's dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question)

My question is what happen to this statement when we are not in $\mathbb{R}^n$ any more, but in a infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question).

What happen to this statement when we are not in $\mathbb{R}^n$ any more, but in an infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Corrected typo in the definition of \tilde{K}
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Loick
Loick
  • 133
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Let $K$ be a closed convex cone in $\mathbb{R}^n$. It's dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\backslash0\}$$\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question)

My question is what happen to this statement when we are not in $\mathbb{R}^n$ any more, but in a infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Let $K$ be a closed convex cone in $\mathbb{R}^n$. It's dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\backslash0\}$. (see for example this question)

My question is what happen to this statement when we are not in $\mathbb{R}^n$ any more, but in a infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

Let $K$ be a closed convex cone in $\mathbb{R}^n$. It's dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$

I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example this question)

My question is what happen to this statement when we are not in $\mathbb{R}^n$ any more, but in a infinite dimensional (non separable) vector space? In fact I am only interested in knowing wether $\tilde{K}$ is included into the interior of $K'$.

I tried to read about weak*-topology and Banach-Alaoglu theorem, but this is really not my cup of tea. Any answer or pointer to suitable literature would be most welcome.

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Loick
Loick
  • 133
  • 4