What's the name of this vector space?

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For a vector space $V$ and its subspace $W$, there is a vector space $W^perp:= xin V^* $.

If $V$ is a finite dimensional inner product vector space, I think $W^perp$ can be called orthogonal complement of $W$. However, what's the name of this in the general case?

edited Jul 23 at 12:47

Rhys Steele

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asked Jul 23 at 12:44

satoukibi

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up vote
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For a vector space $V$ and its subspace $W$, there is a vector space $W^perp:= xin V^* $.

If $V$ is a finite dimensional inner product vector space, I think $W^perp$ can be called orthogonal complement of $W$. However, what's the name of this in the general case?

edited Jul 23 at 12:47

Rhys Steele

5,5751828

asked Jul 23 at 12:44

satoukibi

16016

add a commentÂ |Â

up vote
1
down vote

favorite

For a vector space $V$ and its subspace $W$, there is a vector space $W^perp:= xin V^* $.

If $V$ is a finite dimensional inner product vector space, I think $W^perp$ can be called orthogonal complement of $W$. However, what's the name of this in the general case?

edited Jul 23 at 12:47

Rhys Steele

5,5751828

asked Jul 23 at 12:44

satoukibi

16016

For a vector space $V$ and its subspace $W$, there is a vector space $W^perp:= xin V^* $.

If $V$ is a finite dimensional inner product vector space, I think $W^perp$ can be called orthogonal complement of $W$. However, what's the name of this in the general case?

edited Jul 23 at 12:47

Rhys Steele

5,5751828

asked Jul 23 at 12:44

satoukibi

16016

edited Jul 23 at 12:47

Rhys Steele

5,5751828

edited Jul 23 at 12:47

Rhys Steele

5,5751828

edited Jul 23 at 12:47

Rhys Steele

5,5751828

asked Jul 23 at 12:44

satoukibi

16016

asked Jul 23 at 12:44

satoukibi

16016

asked Jul 23 at 12:44

satoukibi

16016

add a commentÂ |Â

1 Answer
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That's the annihilator of $W$. It applies to any subset of $V$, not just to subspaces.

answered Jul 23 at 12:48

JosÃ© Carlos Santos

113k1698176

3

Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^perp$ and $W' = u in V mid forall y in W, langle u, vrangle = 0 $, which is what most people mean by "$W^perp$". The isomorphism $W' to W^perp$ is simply $u mapsto (v mapsto langle u, v rangle)$; of course, proving it's an isomorphism takes a little bit of work.
â€“Â John Hughes
Jul 23 at 12:54

add a commentÂ |Â

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1 Answer
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1 Answer
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active

oldest

votes

up vote
2
down vote

accepted

That's the annihilator of $W$. It applies to any subset of $V$, not just to subspaces.

answered Jul 23 at 12:48

JosÃ© Carlos Santos

113k1698176

3

Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^perp$ and $W' = u in V mid forall y in W, langle u, vrangle = 0 $, which is what most people mean by "$W^perp$". The isomorphism $W' to W^perp$ is simply $u mapsto (v mapsto langle u, v rangle)$; of course, proving it's an isomorphism takes a little bit of work.
â€“Â John Hughes
Jul 23 at 12:54

add a commentÂ |Â

up vote
2
down vote

accepted

That's the annihilator of $W$. It applies to any subset of $V$, not just to subspaces.

answered Jul 23 at 12:48

113k1698176

3

Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^perp$ and $W' = u in V mid forall y in W, langle u, vrangle = 0 $, which is what most people mean by "$W^perp$". The isomorphism $W' to W^perp$ is simply $u mapsto (v mapsto langle u, v rangle)$; of course, proving it's an isomorphism takes a little bit of work.
â€“Â John Hughes
Jul 23 at 12:54

add a commentÂ |Â

up vote
2
down vote

accepted

That's the annihilator of $W$. It applies to any subset of $V$, not just to subspaces.

answered Jul 23 at 12:48

113k1698176

That's the annihilator of $W$. It applies to any subset of $V$, not just to subspaces.

answered Jul 23 at 12:48

113k1698176

answered Jul 23 at 12:48

113k1698176

answered Jul 23 at 12:48

113k1698176

answered Jul 23 at 12:48

113k1698176

3

Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^perp$ and $W' = u in V mid forall y in W, langle u, vrangle = 0 $, which is what most people mean by "$W^perp$". The isomorphism $W' to W^perp$ is simply $u mapsto (v mapsto langle u, v rangle)$; of course, proving it's an isomorphism takes a little bit of work.
â€“Â John Hughes
Jul 23 at 12:54

add a commentÂ |Â

3

Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^perp$ and $W' = u in V mid forall y in W, langle u, vrangle = 0 $, which is what most people mean by "$W^perp$". The isomorphism $W' to W^perp$ is simply $u mapsto (v mapsto langle u, v rangle)$; of course, proving it's an isomorphism takes a little bit of work.
â€“Â John Hughes
Jul 23 at 12:54

Just to add a little more context: This idea also applies to general vector spaces, $V$, not just those with an inner product. If there is an inner product, and if $W$ is a subspace, then there's a nice isomorphism between the thing OP called $W^perp$ and $W' = u in V mid forall y in W, langle u, vrangle = 0 $, which is what most people mean by "$W^perp$". The isomorphism $W' to W^perp$ is simply $u mapsto (v mapsto langle u, v rangle)$; of course, proving it's an isomorphism takes a little bit of work.
â€“Â John Hughes
Jul 23 at 12:54

add a commentÂ |Â

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