Mixing
Closed unit ball and convex hull of its extreme points
$begingroup$
Let $K$ denote the closed unit ball of $l_1(mathbb{N})$ (considered as a vector space over $mathbb{C}$).
(a) Show that the set of extreme points of $K$ is $mathrm{Ext}(K)= { lambda e_n :lambda in mathbb{C},|lambda|=1,n geq 1}$,
where $(e_n)_{n geq 1}$ is the standard (unit vector) basis for $l_1(mathbb{N})$ (i.e. $e_n$ has $1$ in the $n$-th position and $0$ otherwise).
(b) Show that
$mathrm{co}(mathrm{Ext}(K)) = K ∩ c_c(mathbb{N})$.
(c) Deduce that the closed unit ball of $l_1(mathbb{N})$ is the norm closure of the convex hull
of its extreme points.
(d) Is $mathrm{co}(mathrm{Ext}(K))$ compact in the weak$^∗$-topology on $l_1(mathbb{N})$? Justify your answer.
(Here $l_1(mathbb{N})$ is identified as the dual space of $c_0(mathbb{N})$)
My attempt:
(a) I proved that $lambda e_n$ is $mathrm{Ext}(K)$ for every $n$, but I don't know how to prove that there can't be any other (I would know in the real case, but I'm quite stuck for the complex case).
(b) I proved that if $x$ is in the convex hull of the extreme points of $K$, then it belongs to the intersection, but for the other direction, if $x in K cap c_c(mathbb{N})$ then I get that $||x|| leq 1$. However an element in that convex hull has norm $1$.
(c) I was thinking about using Krein-Milman theorem, but I'm not sure.
(d) By Alaoglu's theorem I know that $K$ is compact in the weak$^*$-topology and $K$ is the norm closure of $mathrm{co}(mathrm{Ext}(K))$. Can I use these facts?
functional-analysis banach-spaces compactness weak-topology
asked Jan 13 at 15:45
user289143user289143
913313
$endgroup$
add a comment |
$begingroup$
Let $K$ denote the closed unit ball of $l_1(mathbb{N})$ (considered as a vector space over $mathbb{C}$).
(a) Show that the set of extreme points of $K$ is $mathrm{Ext}(K)= { lambda e_n :lambda in mathbb{C},|lambda|=1,n geq 1}$,
where $(e_n)_{n geq 1}$ is the standard (unit vector) basis for $l_1(mathbb{N})$ (i.e. $e_n$ has $1$ in the $n$-th position and $0$ otherwise).
(b) Show that
$mathrm{co}(mathrm{Ext}(K)) = K ∩ c_c(mathbb{N})$.
(c) Deduce that the closed unit ball of $l_1(mathbb{N})$ is the norm closure of the convex hull
of its extreme points.
(d) Is $mathrm{co}(mathrm{Ext}(K))$ compact in the weak$^∗$-topology on $l_1(mathbb{N})$? Justify your answer.
(Here $l_1(mathbb{N})$ is identified as the dual space of $c_0(mathbb{N})$)
My attempt:
(a) I proved that $lambda e_n$ is $mathrm{Ext}(K)$ for every $n$, but I don't know how to prove that there can't be any other (I would know in the real case, but I'm quite stuck for the complex case).
(b) I proved that if $x$ is in the convex hull of the extreme points of $K$, then it belongs to the intersection, but for the other direction, if $x in K cap c_c(mathbb{N})$ then I get that $||x|| leq 1$. However an element in that convex hull has norm $1$.
(c) I was thinking about using Krein-Milman theorem, but I'm not sure.
(d) By Alaoglu's theorem I know that $K$ is compact in the weak$^*$-topology and $K$ is the norm closure of $mathrm{co}(mathrm{Ext}(K))$. Can I use these facts?
functional-analysis banach-spaces compactness weak-topology
asked Jan 13 at 15:45
user289143user289143
913313
$endgroup$
add a comment |
$begingroup$
Let $K$ denote the closed unit ball of $l_1(mathbb{N})$ (considered as a vector space over $mathbb{C}$).
(a) Show that the set of extreme points of $K$ is $mathrm{Ext}(K)= { lambda e_n :lambda in mathbb{C},|lambda|=1,n geq 1}$,
where $(e_n)_{n geq 1}$ is the standard (unit vector) basis for $l_1(mathbb{N})$ (i.e. $e_n$ has $1$ in the $n$-th position and $0$ otherwise).
(b) Show that
$mathrm{co}(mathrm{Ext}(K)) = K ∩ c_c(mathbb{N})$.
(c) Deduce that the closed unit ball of $l_1(mathbb{N})$ is the norm closure of the convex hull
of its extreme points.
(d) Is $mathrm{co}(mathrm{Ext}(K))$ compact in the weak$^∗$-topology on $l_1(mathbb{N})$? Justify your answer.
(Here $l_1(mathbb{N})$ is identified as the dual space of $c_0(mathbb{N})$)
My attempt:
(a) I proved that $lambda e_n$ is $mathrm{Ext}(K)$ for every $n$, but I don't know how to prove that there can't be any other (I would know in the real case, but I'm quite stuck for the complex case).
(b) I proved that if $x$ is in the convex hull of the extreme points of $K$, then it belongs to the intersection, but for the other direction, if $x in K cap c_c(mathbb{N})$ then I get that $||x|| leq 1$. However an element in that convex hull has norm $1$.
(c) I was thinking about using Krein-Milman theorem, but I'm not sure.
(d) By Alaoglu's theorem I know that $K$ is compact in the weak$^*$-topology and $K$ is the norm closure of $mathrm{co}(mathrm{Ext}(K))$. Can I use these facts?
functional-analysis banach-spaces compactness weak-topology
asked Jan 13 at 15:45
user289143user289143
913313
$endgroup$
Let $K$ denote the closed unit ball of $l_1(mathbb{N})$ (considered as a vector space over $mathbb{C}$).
(a) Show that the set of extreme points of $K$ is $mathrm{Ext}(K)= { lambda e_n :lambda in mathbb{C},|lambda|=1,n geq 1}$,
where $(e_n)_{n geq 1}$ is the standard (unit vector) basis for $l_1(mathbb{N})$ (i.e. $e_n$ has $1$ in the $n$-th position and $0$ otherwise).
(b) Show that
$mathrm{co}(mathrm{Ext}(K)) = K ∩ c_c(mathbb{N})$.
(c) Deduce that the closed unit ball of $l_1(mathbb{N})$ is the norm closure of the convex hull
of its extreme points.
(d) Is $mathrm{co}(mathrm{Ext}(K))$ compact in the weak$^∗$-topology on $l_1(mathbb{N})$? Justify your answer.
(Here $l_1(mathbb{N})$ is identified as the dual space of $c_0(mathbb{N})$)
My attempt:
(a) I proved that $lambda e_n$ is $mathrm{Ext}(K)$ for every $n$, but I don't know how to prove that there can't be any other (I would know in the real case, but I'm quite stuck for the complex case).
(b) I proved that if $x$ is in the convex hull of the extreme points of $K$, then it belongs to the intersection, but for the other direction, if $x in K cap c_c(mathbb{N})$ then I get that $||x|| leq 1$. However an element in that convex hull has norm $1$.
(c) I was thinking about using Krein-Milman theorem, but I'm not sure.
(d) By Alaoglu's theorem I know that $K$ is compact in the weak$^*$-topology and $K$ is the norm closure of $mathrm{co}(mathrm{Ext}(K))$. Can I use these facts?
functional-analysis banach-spaces compactness weak-topology
functional-analysis banach-spaces compactness weak-topology
asked Jan 13 at 15:45
user289143user289143
913313
asked Jan 13 at 15:45
user289143user289143
913313
asked Jan 13 at 15:45
user289143user289143
913313
asked Jan 13 at 15:45
user289143user289143
913313
asked Jan 13 at 15:45
user289143user289143
913313
913313
add a comment |
add a comment |
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