Mixing
Objects of mixed tensor spaces
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So say I have a real vector space $V$ of dimension $n$. Consider the tensor space $T^(2, 0)(V) = V otimes V$, note that $V otimes V$ is isomorphic to the vector space of multilinear maps from $V^* times V^* to mathbbR$, that is $$T^(2, 0)(V) = V otimes V cong Lleft(V^*, V^*; mathbbRright)$$
So any element $alpha in V otimes V$ is of the form $alpha = mathfraka otimes mathfrakb$ where $mathfraka$ and $mathfrakb$ are elements of the double dual $V^**$ (under an isomorphism to $V$) and $mathfraka otimes mathfrakb : V^* times V^* to mathbbR$ is a multilinear map defined by $(mathfraka otimes mathfrakb)(f, g) = mathfraka(f)cdot mathfrakb(g)$
Now consider the tensor space $T^(0, 2)(V) = V^* otimes V^*$. Similarly we have that $$T^(0, 2)(V) = V^* otimes V^* cong Lleft(V, V; mathbbRright)$$
So any element $beta in V^* otimes V^*$ is of the form $beta = omega otimes eta$ where $omega,eta in V^*$ and $omega otimes eta : V times V to mathbbR$ is a multi-linear map such that $(omega otimes eta)(x, y) = omega(x)cdot eta(y)$.
Firslty is everything above that I've said correct? If so what if we have the mixed tensor space $T^(2, 1)(V) = V otimes V otimes V^*$, is this tensor space isomorphic to $L(V^*, V^*, V; mathbbR)$? In other words is every element $kappa in V otimes V otimes V^*$ of the form $kappa = mathfrakaotimes mathfrakb otimes omega$ where $mathfraka, mathfrakb$ are elements of $V$ under an isomorphism to $V^**$ and $omega in V^*$ and $$kappa = mathfrakaotimes mathfrakb otimes omega : V^* times V^* times V to mathbbR$$ is defined by $$(mathfrakaotimes mathfrakb otimes omega)(f, g, x) = mathfraka(f) cdot mathfrakb(g)cdot omega(x)$$
Is the above correct?
EDIT: Apart from my error in writing everything as simple tensors, are my above ideas correct?
linear-algebra tensor-products tensors multilinear-algebra
asked Aug 6 at 10:18
Perturbative
3,53611039
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1
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So say I have a real vector space $V$ of dimension $n$. Consider the tensor space $T^(2, 0)(V) = V otimes V$, note that $V otimes V$ is isomorphic to the vector space of multilinear maps from $V^* times V^* to mathbbR$, that is $$T^(2, 0)(V) = V otimes V cong Lleft(V^*, V^*; mathbbRright)$$
So any element $alpha in V otimes V$ is of the form $alpha = mathfraka otimes mathfrakb$ where $mathfraka$ and $mathfrakb$ are elements of the double dual $V^**$ (under an isomorphism to $V$) and $mathfraka otimes mathfrakb : V^* times V^* to mathbbR$ is a multilinear map defined by $(mathfraka otimes mathfrakb)(f, g) = mathfraka(f)cdot mathfrakb(g)$
Now consider the tensor space $T^(0, 2)(V) = V^* otimes V^*$. Similarly we have that $$T^(0, 2)(V) = V^* otimes V^* cong Lleft(V, V; mathbbRright)$$
So any element $beta in V^* otimes V^*$ is of the form $beta = omega otimes eta$ where $omega,eta in V^*$ and $omega otimes eta : V times V to mathbbR$ is a multi-linear map such that $(omega otimes eta)(x, y) = omega(x)cdot eta(y)$.
Firslty is everything above that I've said correct? If so what if we have the mixed tensor space $T^(2, 1)(V) = V otimes V otimes V^*$, is this tensor space isomorphic to $L(V^*, V^*, V; mathbbR)$? In other words is every element $kappa in V otimes V otimes V^*$ of the form $kappa = mathfrakaotimes mathfrakb otimes omega$ where $mathfraka, mathfrakb$ are elements of $V$ under an isomorphism to $V^**$ and $omega in V^*$ and $$kappa = mathfrakaotimes mathfrakb otimes omega : V^* times V^* times V to mathbbR$$ is defined by $$(mathfrakaotimes mathfrakb otimes omega)(f, g, x) = mathfraka(f) cdot mathfrakb(g)cdot omega(x)$$
Is the above correct?
EDIT: Apart from my error in writing everything as simple tensors, are my above ideas correct?
linear-algebra tensor-products tensors multilinear-algebra
asked Aug 6 at 10:18
Perturbative
3,53611039
3
Not a complete answer, but for one thing not all tensors are simple; you need $alpha = sum_i mathfraka_i otimes mathfrakb_i$.
– Hans Lundmark
Aug 6 at 10:25
@HansLundmark Apart from that does everything look correct?
– Perturbative
Aug 6 at 10:53
1
You want finite dimensional vector spaces to identify a tensor product with duals of multilinear maps, much like you need it to identify $V$ with its double dual. Apart from this, and the comment on tensors not all being simple, it looks fine.
– Pedro Tamaroff♦
Aug 7 at 6:22
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
So say I have a real vector space $V$ of dimension $n$. Consider the tensor space $T^(2, 0)(V) = V otimes V$, note that $V otimes V$ is isomorphic to the vector space of multilinear maps from $V^* times V^* to mathbbR$, that is $$T^(2, 0)(V) = V otimes V cong Lleft(V^*, V^*; mathbbRright)$$
So any element $alpha in V otimes V$ is of the form $alpha = mathfraka otimes mathfrakb$ where $mathfraka$ and $mathfrakb$ are elements of the double dual $V^**$ (under an isomorphism to $V$) and $mathfraka otimes mathfrakb : V^* times V^* to mathbbR$ is a multilinear map defined by $(mathfraka otimes mathfrakb)(f, g) = mathfraka(f)cdot mathfrakb(g)$
Now consider the tensor space $T^(0, 2)(V) = V^* otimes V^*$. Similarly we have that $$T^(0, 2)(V) = V^* otimes V^* cong Lleft(V, V; mathbbRright)$$
So any element $beta in V^* otimes V^*$ is of the form $beta = omega otimes eta$ where $omega,eta in V^*$ and $omega otimes eta : V times V to mathbbR$ is a multi-linear map such that $(omega otimes eta)(x, y) = omega(x)cdot eta(y)$.
Firslty is everything above that I've said correct? If so what if we have the mixed tensor space $T^(2, 1)(V) = V otimes V otimes V^*$, is this tensor space isomorphic to $L(V^*, V^*, V; mathbbR)$? In other words is every element $kappa in V otimes V otimes V^*$ of the form $kappa = mathfrakaotimes mathfrakb otimes omega$ where $mathfraka, mathfrakb$ are elements of $V$ under an isomorphism to $V^**$ and $omega in V^*$ and $$kappa = mathfrakaotimes mathfrakb otimes omega : V^* times V^* times V to mathbbR$$ is defined by $$(mathfrakaotimes mathfrakb otimes omega)(f, g, x) = mathfraka(f) cdot mathfrakb(g)cdot omega(x)$$
Is the above correct?
EDIT: Apart from my error in writing everything as simple tensors, are my above ideas correct?
linear-algebra tensor-products tensors multilinear-algebra
asked Aug 6 at 10:18
Perturbative
3,53611039
So say I have a real vector space $V$ of dimension $n$. Consider the tensor space $T^(2, 0)(V) = V otimes V$, note that $V otimes V$ is isomorphic to the vector space of multilinear maps from $V^* times V^* to mathbbR$, that is $$T^(2, 0)(V) = V otimes V cong Lleft(V^*, V^*; mathbbRright)$$
So any element $alpha in V otimes V$ is of the form $alpha = mathfraka otimes mathfrakb$ where $mathfraka$ and $mathfrakb$ are elements of the double dual $V^**$ (under an isomorphism to $V$) and $mathfraka otimes mathfrakb : V^* times V^* to mathbbR$ is a multilinear map defined by $(mathfraka otimes mathfrakb)(f, g) = mathfraka(f)cdot mathfrakb(g)$
Now consider the tensor space $T^(0, 2)(V) = V^* otimes V^*$. Similarly we have that $$T^(0, 2)(V) = V^* otimes V^* cong Lleft(V, V; mathbbRright)$$
So any element $beta in V^* otimes V^*$ is of the form $beta = omega otimes eta$ where $omega,eta in V^*$ and $omega otimes eta : V times V to mathbbR$ is a multi-linear map such that $(omega otimes eta)(x, y) = omega(x)cdot eta(y)$.
Firslty is everything above that I've said correct? If so what if we have the mixed tensor space $T^(2, 1)(V) = V otimes V otimes V^*$, is this tensor space isomorphic to $L(V^*, V^*, V; mathbbR)$? In other words is every element $kappa in V otimes V otimes V^*$ of the form $kappa = mathfrakaotimes mathfrakb otimes omega$ where $mathfraka, mathfrakb$ are elements of $V$ under an isomorphism to $V^**$ and $omega in V^*$ and $$kappa = mathfrakaotimes mathfrakb otimes omega : V^* times V^* times V to mathbbR$$ is defined by $$(mathfrakaotimes mathfrakb otimes omega)(f, g, x) = mathfraka(f) cdot mathfrakb(g)cdot omega(x)$$
Is the above correct?
EDIT: Apart from my error in writing everything as simple tensors, are my above ideas correct?
linear-algebra tensor-products tensors multilinear-algebra
asked Aug 6 at 10:18
Perturbative
3,53611039
edited Aug 7 at 6:12
asked Aug 6 at 10:18
Perturbative
3,53611039
asked Aug 6 at 10:18
Perturbative
3,53611039
asked Aug 6 at 10:18
Perturbative
3,53611039
3,53611039
3
Not a complete answer, but for one thing not all tensors are simple; you need $alpha = sum_i mathfraka_i otimes mathfrakb_i$.
– Hans Lundmark
Aug 6 at 10:25
@HansLundmark Apart from that does everything look correct?
– Perturbative
Aug 6 at 10:53
1
You want finite dimensional vector spaces to identify a tensor product with duals of multilinear maps, much like you need it to identify $V$ with its double dual. Apart from this, and the comment on tensors not all being simple, it looks fine.
– Pedro Tamaroff♦
Aug 7 at 6:22
add a comment |Â
3
Not a complete answer, but for one thing not all tensors are simple; you need $alpha = sum_i mathfraka_i otimes mathfrakb_i$.
– Hans Lundmark
Aug 6 at 10:25
@HansLundmark Apart from that does everything look correct?
– Perturbative
Aug 6 at 10:53
1
You want finite dimensional vector spaces to identify a tensor product with duals of multilinear maps, much like you need it to identify $V$ with its double dual. Apart from this, and the comment on tensors not all being simple, it looks fine.
– Pedro Tamaroff♦
Aug 7 at 6:22
3
3
Not a complete answer, but for one thing not all tensors are simple; you need $alpha = sum_i mathfraka_i otimes mathfrakb_i$.
– Hans Lundmark
Aug 6 at 10:25
Not a complete answer, but for one thing not all tensors are simple; you need $alpha = sum_i mathfraka_i otimes mathfrakb_i$.
– Hans Lundmark
Aug 6 at 10:25
@HansLundmark Apart from that does everything look correct?
– Perturbative
Aug 6 at 10:53
@HansLundmark Apart from that does everything look correct?
– Perturbative
Aug 6 at 10:53
1
1
You want finite dimensional vector spaces to identify a tensor product with duals of multilinear maps, much like you need it to identify $V$ with its double dual. Apart from this, and the comment on tensors not all being simple, it looks fine.
– Pedro Tamaroff♦
Aug 7 at 6:22
You want finite dimensional vector spaces to identify a tensor product with duals of multilinear maps, much like you need it to identify $V$ with its double dual. Apart from this, and the comment on tensors not all being simple, it looks fine.
– Pedro Tamaroff♦
Aug 7 at 6:22
add a comment |Â
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3
Not a complete answer, but for one thing not all tensors are simple; you need $alpha = sum_i mathfraka_i otimes mathfrakb_i$.
– Hans Lundmark
Aug 6 at 10:25
@HansLundmark Apart from that does everything look correct?
– Perturbative
Aug 6 at 10:53
1
You want finite dimensional vector spaces to identify a tensor product with duals of multilinear maps, much like you need it to identify $V$ with its double dual. Apart from this, and the comment on tensors not all being simple, it looks fine.
– Pedro Tamaroff♦
Aug 7 at 6:22