$(V, geq)$ Banach lattice, $(W, geq)$ Riesz space, then $mathcal{L}(V, W)^+ = mathcal{B}(V, W)^+$
Let $(V, geq)$ be a Banach lattice and $(W, geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $mathcal{L}(V, W)^+ = mathcal{B}(V, W)^+$, where $mathcal{L}(V, W)^+$, $mathcal{B}(V, W)^+$ are the sets of positive elements of $mathcal{L}(V, W)$ and $mathcal{B}(V, W)$, respectevely.
Trivially, $mathcal{B}(V, W)^+ subset mathcal{L}(V, W)^+$. So it remains to show that $mathcal{L}(V, W)^+ subset mathcal{B}(V, W)^+$, which means that every positive linear functional between $V$ and $W$ is bounded.
Let $fcolon V rightarrow W in mathcal{L}(V, W)^+$.Thus, by definition, $f(v) geq 0$ if $v geq 0$.
So, for $v geq 0$, we have that $f(v) in W^+$, which is generated by some positive element of $W$. Formally, there is $w_0$ such that $W^+ = cup_{alpha in mathbb{R}^+_0} [0, alpha w_0]$.
Then, there is $beta in mathbb{R}^+_0$ such that $f(v) = beta w_0$ (is this correct?).
Now I'm having some trouble to conclude that $f$ is bounded.
I would appreciate if someone could help me!
banach-spaces lattice-orders banach-lattices
asked Nov 24 at 17:38
Luísa Borsato
1,496315
add a comment |
Let $(V, geq)$ be a Banach lattice and $(W, geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $mathcal{L}(V, W)^+ = mathcal{B}(V, W)^+$, where $mathcal{L}(V, W)^+$, $mathcal{B}(V, W)^+$ are the sets of positive elements of $mathcal{L}(V, W)$ and $mathcal{B}(V, W)$, respectevely.
Trivially, $mathcal{B}(V, W)^+ subset mathcal{L}(V, W)^+$. So it remains to show that $mathcal{L}(V, W)^+ subset mathcal{B}(V, W)^+$, which means that every positive linear functional between $V$ and $W$ is bounded.
Let $fcolon V rightarrow W in mathcal{L}(V, W)^+$.Thus, by definition, $f(v) geq 0$ if $v geq 0$.
So, for $v geq 0$, we have that $f(v) in W^+$, which is generated by some positive element of $W$. Formally, there is $w_0$ such that $W^+ = cup_{alpha in mathbb{R}^+_0} [0, alpha w_0]$.
Then, there is $beta in mathbb{R}^+_0$ such that $f(v) = beta w_0$ (is this correct?).
Now I'm having some trouble to conclude that $f$ is bounded.
I would appreciate if someone could help me!
banach-spaces lattice-orders banach-lattices
asked Nov 24 at 17:38
Luísa Borsato
1,496315
add a comment |
Let $(V, geq)$ be a Banach lattice and $(W, geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $mathcal{L}(V, W)^+ = mathcal{B}(V, W)^+$, where $mathcal{L}(V, W)^+$, $mathcal{B}(V, W)^+$ are the sets of positive elements of $mathcal{L}(V, W)$ and $mathcal{B}(V, W)$, respectevely.
Trivially, $mathcal{B}(V, W)^+ subset mathcal{L}(V, W)^+$. So it remains to show that $mathcal{L}(V, W)^+ subset mathcal{B}(V, W)^+$, which means that every positive linear functional between $V$ and $W$ is bounded.
Let $fcolon V rightarrow W in mathcal{L}(V, W)^+$.Thus, by definition, $f(v) geq 0$ if $v geq 0$.
So, for $v geq 0$, we have that $f(v) in W^+$, which is generated by some positive element of $W$. Formally, there is $w_0$ such that $W^+ = cup_{alpha in mathbb{R}^+_0} [0, alpha w_0]$.
Then, there is $beta in mathbb{R}^+_0$ such that $f(v) = beta w_0$ (is this correct?).
Now I'm having some trouble to conclude that $f$ is bounded.
I would appreciate if someone could help me!
banach-spaces lattice-orders banach-lattices
asked Nov 24 at 17:38
Luísa Borsato
1,496315
Let $(V, geq)$ be a Banach lattice and $(W, geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $mathcal{L}(V, W)^+ = mathcal{B}(V, W)^+$, where $mathcal{L}(V, W)^+$, $mathcal{B}(V, W)^+$ are the sets of positive elements of $mathcal{L}(V, W)$ and $mathcal{B}(V, W)$, respectevely.
Trivially, $mathcal{B}(V, W)^+ subset mathcal{L}(V, W)^+$. So it remains to show that $mathcal{L}(V, W)^+ subset mathcal{B}(V, W)^+$, which means that every positive linear functional between $V$ and $W$ is bounded.
Let $fcolon V rightarrow W in mathcal{L}(V, W)^+$.Thus, by definition, $f(v) geq 0$ if $v geq 0$.
So, for $v geq 0$, we have that $f(v) in W^+$, which is generated by some positive element of $W$. Formally, there is $w_0$ such that $W^+ = cup_{alpha in mathbb{R}^+_0} [0, alpha w_0]$.
Then, there is $beta in mathbb{R}^+_0$ such that $f(v) = beta w_0$ (is this correct?).
Now I'm having some trouble to conclude that $f$ is bounded.
I would appreciate if someone could help me!
banach-spaces lattice-orders banach-lattices
banach-spaces lattice-orders banach-lattices
asked Nov 24 at 17:38
Luísa Borsato
1,496315
asked Nov 24 at 17:38
Luísa Borsato
1,496315
asked Nov 24 at 17:38
Luísa Borsato
1,496315
asked Nov 24 at 17:38
Luísa Borsato
1,496315
asked Nov 24 at 17:38
Luísa Borsato
1,496315
1,496315
add a comment |
add a comment |
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