Are antilinear forms part of the tensor algebra of a $mathbb{C}$-vector field?
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Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $mathcal{T}(V)$ be its tensor algebra. If $K$ is either $mathbb{R}$ or $mathbb{C}$, then every multilinear form
$$Vtimescdotstimes V times V^*timescdotstimes V^* longrightarrow K$$
can be canonically identified with one element of $mathcal{T}(V)$ (and one element only).
If we consider $K=mathbb{C}$, though, no element of $mathcal{T}(V)$ can be identified with an antilinear operator, sesquilinear form or any application that's antilinear with respect to some variables (and linear for the rest of them), right?
If so, can the tensor algebra of $V$ be extended so that it also contains tensors canonically identifiable with those?, how?
complex-numbers tensor-products multilinear-algebra
asked Nov 16 at 20:35
TeicDaun
7116
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up vote
1
down vote
favorite
Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $mathcal{T}(V)$ be its tensor algebra. If $K$ is either $mathbb{R}$ or $mathbb{C}$, then every multilinear form
$$Vtimescdotstimes V times V^*timescdotstimes V^* longrightarrow K$$
can be canonically identified with one element of $mathcal{T}(V)$ (and one element only).
If we consider $K=mathbb{C}$, though, no element of $mathcal{T}(V)$ can be identified with an antilinear operator, sesquilinear form or any application that's antilinear with respect to some variables (and linear for the rest of them), right?
If so, can the tensor algebra of $V$ be extended so that it also contains tensors canonically identifiable with those?, how?
complex-numbers tensor-products multilinear-algebra
asked Nov 16 at 20:35
TeicDaun
7116
Maybe something semilinear.
– Ricardo Buring
Nov 18 at 22:38
If you have a scalar product you can identify $V$ with the space of anti-linear forms. Alternatively call $overline{V}$ the "anti-linear dual" and note that anti-linear forms are of the form $overline{V}times..times overline{V}times overline{V}^*times...timesoverline V^*to K$.
– s.harp
Nov 21 at 10:27
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $mathcal{T}(V)$ be its tensor algebra. If $K$ is either $mathbb{R}$ or $mathbb{C}$, then every multilinear form
$$Vtimescdotstimes V times V^*timescdotstimes V^* longrightarrow K$$
can be canonically identified with one element of $mathcal{T}(V)$ (and one element only).
If we consider $K=mathbb{C}$, though, no element of $mathcal{T}(V)$ can be identified with an antilinear operator, sesquilinear form or any application that's antilinear with respect to some variables (and linear for the rest of them), right?
If so, can the tensor algebra of $V$ be extended so that it also contains tensors canonically identifiable with those?, how?
complex-numbers tensor-products multilinear-algebra
asked Nov 16 at 20:35
TeicDaun
7116
Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $mathcal{T}(V)$ be its tensor algebra. If $K$ is either $mathbb{R}$ or $mathbb{C}$, then every multilinear form
$$Vtimescdotstimes V times V^*timescdotstimes V^* longrightarrow K$$
can be canonically identified with one element of $mathcal{T}(V)$ (and one element only).
If we consider $K=mathbb{C}$, though, no element of $mathcal{T}(V)$ can be identified with an antilinear operator, sesquilinear form or any application that's antilinear with respect to some variables (and linear for the rest of them), right?
If so, can the tensor algebra of $V$ be extended so that it also contains tensors canonically identifiable with those?, how?
complex-numbers tensor-products multilinear-algebra
complex-numbers tensor-products multilinear-algebra
asked Nov 16 at 20:35
TeicDaun
7116
asked Nov 16 at 20:35
TeicDaun
7116
asked Nov 16 at 20:35
TeicDaun
7116
asked Nov 16 at 20:35
TeicDaun
7116
asked Nov 16 at 20:35
TeicDaun
7116
7116
Maybe something semilinear.
– Ricardo Buring
Nov 18 at 22:38
If you have a scalar product you can identify $V$ with the space of anti-linear forms. Alternatively call $overline{V}$ the "anti-linear dual" and note that anti-linear forms are of the form $overline{V}times..times overline{V}times overline{V}^*times...timesoverline V^*to K$.
– s.harp
Nov 21 at 10:27
add a comment |
Maybe something semilinear.
– Ricardo Buring
Nov 18 at 22:38
If you have a scalar product you can identify $V$ with the space of anti-linear forms. Alternatively call $overline{V}$ the "anti-linear dual" and note that anti-linear forms are of the form $overline{V}times..times overline{V}times overline{V}^*times...timesoverline V^*to K$.
– s.harp
Nov 21 at 10:27
Maybe something semilinear.
– Ricardo Buring
Nov 18 at 22:38
Maybe something semilinear.
– Ricardo Buring
Nov 18 at 22:38
If you have a scalar product you can identify $V$ with the space of anti-linear forms. Alternatively call $overline{V}$ the "anti-linear dual" and note that anti-linear forms are of the form $overline{V}times..times overline{V}times overline{V}^*times...timesoverline V^*to K$.
– s.harp
Nov 21 at 10:27
If you have a scalar product you can identify $V$ with the space of anti-linear forms. Alternatively call $overline{V}$ the "anti-linear dual" and note that anti-linear forms are of the form $overline{V}times..times overline{V}times overline{V}^*times...timesoverline V^*to K$.
– s.harp
Nov 21 at 10:27
add a comment |
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Maybe something semilinear.
– Ricardo Buring
Nov 18 at 22:38
If you have a scalar product you can identify $V$ with the space of anti-linear forms. Alternatively call $overline{V}$ the "anti-linear dual" and note that anti-linear forms are of the form $overline{V}times..times overline{V}times overline{V}^*times...timesoverline V^*to K$.
– s.harp
Nov 21 at 10:27