An Isomorphism for the Tensor Product in Infinite Dimensions
$begingroup$
For finite-dimensional vector spaces $V$ and $W$, we have that $V cong V^{**}$ and that $V^* otimes W cong text{Hom}(V,W)$. We can then combine these two isomorphisms to get
$$
(1) V otimes W cong V^{**} otimes W cong text{Hom}(V^*, W)
$$
Now, let $V$ and $W$ be two infinite-dimensional vector spaces, and define $text{Hom}_{ fin}(V,W)$ to be the finite-rank homomorphisms from $V$ to $W$. I know that we have an isomorphism $V^* otimes W cong text{Hom}_{ fin}(V,W)$.
Question: Can we recover a result that generalizes $(1)$? For example, do we have $V otimes W cong text{Hom}_{ fin}(V^*,W)$?
I believe the map $mu: V otimes W to text{Hom}_{ fin}(V^*,W)$ defined by
$$
v otimes w to (f to f(v)w)
$$
gives an injection; however, I have doubts about this map being a surjection. I have a feeling that surjectivity would rely on $V cong V^{**}$, which we don't have in the infinite-dimensional case.
linear-algebra tensor-products
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
$endgroup$
add a comment |
$begingroup$
For finite-dimensional vector spaces $V$ and $W$, we have that $V cong V^{**}$ and that $V^* otimes W cong text{Hom}(V,W)$. We can then combine these two isomorphisms to get
$$
(1) V otimes W cong V^{**} otimes W cong text{Hom}(V^*, W)
$$
Now, let $V$ and $W$ be two infinite-dimensional vector spaces, and define $text{Hom}_{ fin}(V,W)$ to be the finite-rank homomorphisms from $V$ to $W$. I know that we have an isomorphism $V^* otimes W cong text{Hom}_{ fin}(V,W)$.
Question: Can we recover a result that generalizes $(1)$? For example, do we have $V otimes W cong text{Hom}_{ fin}(V^*,W)$?
I believe the map $mu: V otimes W to text{Hom}_{ fin}(V^*,W)$ defined by
$$
v otimes w to (f to f(v)w)
$$
gives an injection; however, I have doubts about this map being a surjection. I have a feeling that surjectivity would rely on $V cong V^{**}$, which we don't have in the infinite-dimensional case.
linear-algebra tensor-products
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
$endgroup$
$begingroup$
It should probably work if you take $hom_{mathrm {Fin}}(V,mathbb k)$ as $V^*$, but in general the right hand side is much larger than $Votimes W$.
$endgroup$
– Pedro Tamaroff♦
Dec 25 '18 at 1:26
$begingroup$
In the case I'm interested in, I have a grading $V = bigoplus_lambda V_lambda$ where $text{dim}(V_lambda)<infty$. Then if we set $$ V^vee := bigoplus_lambda V^*_lambda $$ the "graded" dual of $V$, I believe (although I could be mistaken) that $V^vee = text{Hom}_{ fin}(V,mathbb{k})$. If so, you are suggesting that one could possibly instead consider $text{Hom}_{ fin}(V^vee,W)$? My intuition and knowledge of infinite-dimensional linear algebra is embarrassingly lacking, so I would have to work through the details.
$endgroup$
– SamJeralds
Dec 25 '18 at 3:32
add a comment |
$begingroup$
For finite-dimensional vector spaces $V$ and $W$, we have that $V cong V^{**}$ and that $V^* otimes W cong text{Hom}(V,W)$. We can then combine these two isomorphisms to get
$$
(1) V otimes W cong V^{**} otimes W cong text{Hom}(V^*, W)
$$
Now, let $V$ and $W$ be two infinite-dimensional vector spaces, and define $text{Hom}_{ fin}(V,W)$ to be the finite-rank homomorphisms from $V$ to $W$. I know that we have an isomorphism $V^* otimes W cong text{Hom}_{ fin}(V,W)$.
Question: Can we recover a result that generalizes $(1)$? For example, do we have $V otimes W cong text{Hom}_{ fin}(V^*,W)$?
I believe the map $mu: V otimes W to text{Hom}_{ fin}(V^*,W)$ defined by
$$
v otimes w to (f to f(v)w)
$$
gives an injection; however, I have doubts about this map being a surjection. I have a feeling that surjectivity would rely on $V cong V^{**}$, which we don't have in the infinite-dimensional case.
linear-algebra tensor-products
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
$endgroup$
For finite-dimensional vector spaces $V$ and $W$, we have that $V cong V^{**}$ and that $V^* otimes W cong text{Hom}(V,W)$. We can then combine these two isomorphisms to get
$$
(1) V otimes W cong V^{**} otimes W cong text{Hom}(V^*, W)
$$
Now, let $V$ and $W$ be two infinite-dimensional vector spaces, and define $text{Hom}_{ fin}(V,W)$ to be the finite-rank homomorphisms from $V$ to $W$. I know that we have an isomorphism $V^* otimes W cong text{Hom}_{ fin}(V,W)$.
Question: Can we recover a result that generalizes $(1)$? For example, do we have $V otimes W cong text{Hom}_{ fin}(V^*,W)$?
I believe the map $mu: V otimes W to text{Hom}_{ fin}(V^*,W)$ defined by
$$
v otimes w to (f to f(v)w)
$$
gives an injection; however, I have doubts about this map being a surjection. I have a feeling that surjectivity would rely on $V cong V^{**}$, which we don't have in the infinite-dimensional case.
linear-algebra tensor-products
linear-algebra tensor-products
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
asked Dec 25 '18 at 1:14
SamJeraldsSamJeralds
261
261
$begingroup$
It should probably work if you take $hom_{mathrm {Fin}}(V,mathbb k)$ as $V^*$, but in general the right hand side is much larger than $Votimes W$.
$endgroup$
– Pedro Tamaroff♦
Dec 25 '18 at 1:26
$begingroup$
In the case I'm interested in, I have a grading $V = bigoplus_lambda V_lambda$ where $text{dim}(V_lambda)<infty$. Then if we set $$ V^vee := bigoplus_lambda V^*_lambda $$ the "graded" dual of $V$, I believe (although I could be mistaken) that $V^vee = text{Hom}_{ fin}(V,mathbb{k})$. If so, you are suggesting that one could possibly instead consider $text{Hom}_{ fin}(V^vee,W)$? My intuition and knowledge of infinite-dimensional linear algebra is embarrassingly lacking, so I would have to work through the details.
$endgroup$
– SamJeralds
Dec 25 '18 at 3:32
add a comment |
$begingroup$
It should probably work if you take $hom_{mathrm {Fin}}(V,mathbb k)$ as $V^*$, but in general the right hand side is much larger than $Votimes W$.
$endgroup$
– Pedro Tamaroff♦
Dec 25 '18 at 1:26
$begingroup$
In the case I'm interested in, I have a grading $V = bigoplus_lambda V_lambda$ where $text{dim}(V_lambda)<infty$. Then if we set $$ V^vee := bigoplus_lambda V^*_lambda $$ the "graded" dual of $V$, I believe (although I could be mistaken) that $V^vee = text{Hom}_{ fin}(V,mathbb{k})$. If so, you are suggesting that one could possibly instead consider $text{Hom}_{ fin}(V^vee,W)$? My intuition and knowledge of infinite-dimensional linear algebra is embarrassingly lacking, so I would have to work through the details.
$endgroup$
– SamJeralds
Dec 25 '18 at 3:32
$begingroup$
It should probably work if you take $hom_{mathrm {Fin}}(V,mathbb k)$ as $V^*$, but in general the right hand side is much larger than $Votimes W$.
$endgroup$
– Pedro Tamaroff♦
Dec 25 '18 at 1:26
$begingroup$
It should probably work if you take $hom_{mathrm {Fin}}(V,mathbb k)$ as $V^*$, but in general the right hand side is much larger than $Votimes W$.
$endgroup$
– Pedro Tamaroff♦
Dec 25 '18 at 1:26
$begingroup$
In the case I'm interested in, I have a grading $V = bigoplus_lambda V_lambda$ where $text{dim}(V_lambda)<infty$. Then if we set $$ V^vee := bigoplus_lambda V^*_lambda $$ the "graded" dual of $V$, I believe (although I could be mistaken) that $V^vee = text{Hom}_{ fin}(V,mathbb{k})$. If so, you are suggesting that one could possibly instead consider $text{Hom}_{ fin}(V^vee,W)$? My intuition and knowledge of infinite-dimensional linear algebra is embarrassingly lacking, so I would have to work through the details.
$endgroup$
– SamJeralds
Dec 25 '18 at 3:32
$begingroup$
In the case I'm interested in, I have a grading $V = bigoplus_lambda V_lambda$ where $text{dim}(V_lambda)<infty$. Then if we set $$ V^vee := bigoplus_lambda V^*_lambda $$ the "graded" dual of $V$, I believe (although I could be mistaken) that $V^vee = text{Hom}_{ fin}(V,mathbb{k})$. If so, you are suggesting that one could possibly instead consider $text{Hom}_{ fin}(V^vee,W)$? My intuition and knowledge of infinite-dimensional linear algebra is embarrassingly lacking, so I would have to work through the details.
$endgroup$
– SamJeralds
Dec 25 '18 at 3:32
add a comment |
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$begingroup$
It should probably work if you take $hom_{mathrm {Fin}}(V,mathbb k)$ as $V^*$, but in general the right hand side is much larger than $Votimes W$.
$endgroup$
– Pedro Tamaroff♦
Dec 25 '18 at 1:26
$begingroup$
In the case I'm interested in, I have a grading $V = bigoplus_lambda V_lambda$ where $text{dim}(V_lambda)<infty$. Then if we set $$ V^vee := bigoplus_lambda V^*_lambda $$ the "graded" dual of $V$, I believe (although I could be mistaken) that $V^vee = text{Hom}_{ fin}(V,mathbb{k})$. If so, you are suggesting that one could possibly instead consider $text{Hom}_{ fin}(V^vee,W)$? My intuition and knowledge of infinite-dimensional linear algebra is embarrassingly lacking, so I would have to work through the details.
$endgroup$
– SamJeralds
Dec 25 '18 at 3:32