A closed convex set in a complex Banach space.
up vote
1
down vote
favorite
Let $X$ be a Banach space over the field of complex numbers. Let $K$ be a subset in $X$ which contains all linear combinations $Z_1x+Z_2y$ for all $Z_1,Z_2in mathbb C$ with $Re(Z_1)geq0$ & $Re(Z_2)geq0$ and for all $x,y in K$.
I want to prove or disprove that $K$ is closed convex in $X$. I have proved $K$ is convex, but I am stuck to prove or disprove that $K$ is closed.
functional-analysis banach-spaces normed-spaces
asked Nov 21 at 11:03
Infinity
543313
add a comment |
up vote
1
down vote
favorite
Let $X$ be a Banach space over the field of complex numbers. Let $K$ be a subset in $X$ which contains all linear combinations $Z_1x+Z_2y$ for all $Z_1,Z_2in mathbb C$ with $Re(Z_1)geq0$ & $Re(Z_2)geq0$ and for all $x,y in K$.
I want to prove or disprove that $K$ is closed convex in $X$. I have proved $K$ is convex, but I am stuck to prove or disprove that $K$ is closed.
functional-analysis banach-spaces normed-spaces
asked Nov 21 at 11:03
Infinity
543313
Think about an open angle or cone..
– Berci
Nov 21 at 11:23
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $X$ be a Banach space over the field of complex numbers. Let $K$ be a subset in $X$ which contains all linear combinations $Z_1x+Z_2y$ for all $Z_1,Z_2in mathbb C$ with $Re(Z_1)geq0$ & $Re(Z_2)geq0$ and for all $x,y in K$.
I want to prove or disprove that $K$ is closed convex in $X$. I have proved $K$ is convex, but I am stuck to prove or disprove that $K$ is closed.
functional-analysis banach-spaces normed-spaces
asked Nov 21 at 11:03
Infinity
543313
Let $X$ be a Banach space over the field of complex numbers. Let $K$ be a subset in $X$ which contains all linear combinations $Z_1x+Z_2y$ for all $Z_1,Z_2in mathbb C$ with $Re(Z_1)geq0$ & $Re(Z_2)geq0$ and for all $x,y in K$.
I want to prove or disprove that $K$ is closed convex in $X$. I have proved $K$ is convex, but I am stuck to prove or disprove that $K$ is closed.
functional-analysis banach-spaces normed-spaces
functional-analysis banach-spaces normed-spaces
asked Nov 21 at 11:03
Infinity
543313
asked Nov 21 at 11:03
Infinity
543313
asked Nov 21 at 11:03
Infinity
543313
asked Nov 21 at 11:03
Infinity
543313
asked Nov 21 at 11:03
Infinity
543313
543313
Think about an open angle or cone..
– Berci
Nov 21 at 11:23
add a comment |
Think about an open angle or cone..
– Berci
Nov 21 at 11:23
Think about an open angle or cone..
– Berci
Nov 21 at 11:23
Think about an open angle or cone..
– Berci
Nov 21 at 11:23
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
Not every linear subspace of a Banach space is closed so the answer is NO. [ Eg. $X=l^{1}$, $K$ is the space of all sequences with only finite number of non-zero entries].
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Not every linear subspace of a Banach space is closed so the answer is NO. [ Eg. $X=l^{1}$, $K$ is the space of all sequences with only finite number of non-zero entries].
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
add a comment |
up vote
3
down vote
Not every linear subspace of a Banach space is closed so the answer is NO. [ Eg. $X=l^{1}$, $K$ is the space of all sequences with only finite number of non-zero entries].
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
add a comment |
up vote
3
down vote
up vote
3
down vote
Not every linear subspace of a Banach space is closed so the answer is NO. [ Eg. $X=l^{1}$, $K$ is the space of all sequences with only finite number of non-zero entries].
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
Not every linear subspace of a Banach space is closed so the answer is NO. [ Eg. $X=l^{1}$, $K$ is the space of all sequences with only finite number of non-zero entries].
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
answered Nov 21 at 12:05
Kavi Rama Murthy
42.4k31751
42.4k31751
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
add a comment |
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
But K in my question is not a subspace.
– Infinity
Nov 21 at 13:56
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
@Infinity Any linear subspace satisfies your conditions on $K$.
– Kavi Rama Murthy
Nov 21 at 23:09
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007567%2fa-closed-convex-set-in-a-complex-banach-space%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Think about an open angle or cone..
– Berci
Nov 21 at 11:23