convex cone in complex Banach space
up vote
0
down vote
favorite
A convex cone is defined as (by Wikipedia):
A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.
In my research work, I need a convex cone in a complex Banach space, but the set of complex numbers is not an ordered field. Then how to define a convex cone in a complex Banach space? I tried to define such a partial order on $mathbb C$ so that it can be a total order on $mathbb C$ but I could not succeed.
functional-analysis banach-spaces normed-spaces
asked Nov 22 at 9:32
Infinity
545313
add a comment |
up vote
0
down vote
favorite
A convex cone is defined as (by Wikipedia):
A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.
In my research work, I need a convex cone in a complex Banach space, but the set of complex numbers is not an ordered field. Then how to define a convex cone in a complex Banach space? I tried to define such a partial order on $mathbb C$ so that it can be a total order on $mathbb C$ but I could not succeed.
functional-analysis banach-spaces normed-spaces
asked Nov 22 at 9:32
Infinity
545313
1
Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces.
– Kavi Rama Murthy
Nov 22 at 9:34
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A convex cone is defined as (by Wikipedia):
A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.
In my research work, I need a convex cone in a complex Banach space, but the set of complex numbers is not an ordered field. Then how to define a convex cone in a complex Banach space? I tried to define such a partial order on $mathbb C$ so that it can be a total order on $mathbb C$ but I could not succeed.
functional-analysis banach-spaces normed-spaces
asked Nov 22 at 9:32
Infinity
545313
A convex cone is defined as (by Wikipedia):
A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.
In my research work, I need a convex cone in a complex Banach space, but the set of complex numbers is not an ordered field. Then how to define a convex cone in a complex Banach space? I tried to define such a partial order on $mathbb C$ so that it can be a total order on $mathbb C$ but I could not succeed.
functional-analysis banach-spaces normed-spaces
functional-analysis banach-spaces normed-spaces
asked Nov 22 at 9:32
Infinity
545313
asked Nov 22 at 9:32
Infinity
545313
asked Nov 22 at 9:32
Infinity
545313
asked Nov 22 at 9:32
Infinity
545313
asked Nov 22 at 9:32
Infinity
545313
545313
1
Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces.
– Kavi Rama Murthy
Nov 22 at 9:34
add a comment |
1
Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces.
– Kavi Rama Murthy
Nov 22 at 9:34
1
1
Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces.
– Kavi Rama Murthy
Nov 22 at 9:34
Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces.
– Kavi Rama Murthy
Nov 22 at 9:34
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
If $X$ is a complex Banach space and $C subseteq X$, then $C$ is called a convex cone if $x,y in C $ and $s,t in [0, infty)$ imply that $sx+ty in C$.
answered Nov 22 at 9:59
Fred
42.4k1642
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
If $X$ is a complex Banach space and $C subseteq X$, then $C$ is called a convex cone if $x,y in C $ and $s,t in [0, infty)$ imply that $sx+ty in C$.
answered Nov 22 at 9:59
Fred
42.4k1642
add a comment |
up vote
0
down vote
accepted
If $X$ is a complex Banach space and $C subseteq X$, then $C$ is called a convex cone if $x,y in C $ and $s,t in [0, infty)$ imply that $sx+ty in C$.
answered Nov 22 at 9:59
Fred
42.4k1642
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
If $X$ is a complex Banach space and $C subseteq X$, then $C$ is called a convex cone if $x,y in C $ and $s,t in [0, infty)$ imply that $sx+ty in C$.
answered Nov 22 at 9:59
Fred
42.4k1642
If $X$ is a complex Banach space and $C subseteq X$, then $C$ is called a convex cone if $x,y in C $ and $s,t in [0, infty)$ imply that $sx+ty in C$.
answered Nov 22 at 9:59
Fred
42.4k1642
answered Nov 22 at 9:59
Fred
42.4k1642
answered Nov 22 at 9:59
Fred
42.4k1642
answered Nov 22 at 9:59
Fred
42.4k1642
42.4k1642
add a comment |
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%2f3008912%2fconvex-cone-in-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
1
Convex cones are defined the same for complex Banach spaces as they are defined for real Banach spaces.
– Kavi Rama Murthy
Nov 22 at 9:34