Is there a half-dual space?
Assume we have a finite dimensional vector space $V$ and its dual $V^*$. Assume we are also given a non-degenerate inner product $Phi$ on $V$. The inner product induces an inner product on the dual, which can be denoted s the inverse $Phi^*$ and would correspond to the inverse matrix if we chose basis. My question is is there any mathematical notion of "half-dual" space or any space that resides in the middle. To make this a bit more clear, my line of thoughts are:
The inner product $Phi$ is a map from $V times V tilde{to} mathbb{R}$, however, it can also be looked at as $V tilde{to} V^*$.
If it is not degenerate the dual metric is defined as $Phi^*(u^*, v^*) = Phi(Phi^{-1}(u^*), Phi^{-1}(v^*))$
If we can define two maps $phi_1 : V tilde{to} tilde{V}$ and $phi_2 : tilde{V} tilde{to} V^*$ such that $phi_2 circ phi_1 = Phi$ and both invertible.
The inner product on $tilde{V}$ to be similarly inherited from $Phi$.
Now my background is not mathematics so what I'm looking for is after choosing a basis $Phi$ is a matrix and the map can be looked as $v^* = Phi v$. The question is is there any notion or more rigorous definition, that is independent of the choice of basis that corresponds to $Phi^frac{1}{2} v$? Or is this just something that exists only in the chosen basis?
linear-algebra vector-spaces inner-product-space
asked Nov 28 '18 at 2:29
Alex Botev
674314
add a comment |
Assume we have a finite dimensional vector space $V$ and its dual $V^*$. Assume we are also given a non-degenerate inner product $Phi$ on $V$. The inner product induces an inner product on the dual, which can be denoted s the inverse $Phi^*$ and would correspond to the inverse matrix if we chose basis. My question is is there any mathematical notion of "half-dual" space or any space that resides in the middle. To make this a bit more clear, my line of thoughts are:
The inner product $Phi$ is a map from $V times V tilde{to} mathbb{R}$, however, it can also be looked at as $V tilde{to} V^*$.
If it is not degenerate the dual metric is defined as $Phi^*(u^*, v^*) = Phi(Phi^{-1}(u^*), Phi^{-1}(v^*))$
If we can define two maps $phi_1 : V tilde{to} tilde{V}$ and $phi_2 : tilde{V} tilde{to} V^*$ such that $phi_2 circ phi_1 = Phi$ and both invertible.
The inner product on $tilde{V}$ to be similarly inherited from $Phi$.
Now my background is not mathematics so what I'm looking for is after choosing a basis $Phi$ is a matrix and the map can be looked as $v^* = Phi v$. The question is is there any notion or more rigorous definition, that is independent of the choice of basis that corresponds to $Phi^frac{1}{2} v$? Or is this just something that exists only in the chosen basis?
linear-algebra vector-spaces inner-product-space
asked Nov 28 '18 at 2:29
Alex Botev
674314
add a comment |
Assume we have a finite dimensional vector space $V$ and its dual $V^*$. Assume we are also given a non-degenerate inner product $Phi$ on $V$. The inner product induces an inner product on the dual, which can be denoted s the inverse $Phi^*$ and would correspond to the inverse matrix if we chose basis. My question is is there any mathematical notion of "half-dual" space or any space that resides in the middle. To make this a bit more clear, my line of thoughts are:
The inner product $Phi$ is a map from $V times V tilde{to} mathbb{R}$, however, it can also be looked at as $V tilde{to} V^*$.
If it is not degenerate the dual metric is defined as $Phi^*(u^*, v^*) = Phi(Phi^{-1}(u^*), Phi^{-1}(v^*))$
If we can define two maps $phi_1 : V tilde{to} tilde{V}$ and $phi_2 : tilde{V} tilde{to} V^*$ such that $phi_2 circ phi_1 = Phi$ and both invertible.
The inner product on $tilde{V}$ to be similarly inherited from $Phi$.
Now my background is not mathematics so what I'm looking for is after choosing a basis $Phi$ is a matrix and the map can be looked as $v^* = Phi v$. The question is is there any notion or more rigorous definition, that is independent of the choice of basis that corresponds to $Phi^frac{1}{2} v$? Or is this just something that exists only in the chosen basis?
linear-algebra vector-spaces inner-product-space
asked Nov 28 '18 at 2:29
Alex Botev
674314
Assume we have a finite dimensional vector space $V$ and its dual $V^*$. Assume we are also given a non-degenerate inner product $Phi$ on $V$. The inner product induces an inner product on the dual, which can be denoted s the inverse $Phi^*$ and would correspond to the inverse matrix if we chose basis. My question is is there any mathematical notion of "half-dual" space or any space that resides in the middle. To make this a bit more clear, my line of thoughts are:
The inner product $Phi$ is a map from $V times V tilde{to} mathbb{R}$, however, it can also be looked at as $V tilde{to} V^*$.
If it is not degenerate the dual metric is defined as $Phi^*(u^*, v^*) = Phi(Phi^{-1}(u^*), Phi^{-1}(v^*))$
If we can define two maps $phi_1 : V tilde{to} tilde{V}$ and $phi_2 : tilde{V} tilde{to} V^*$ such that $phi_2 circ phi_1 = Phi$ and both invertible.
The inner product on $tilde{V}$ to be similarly inherited from $Phi$.
Now my background is not mathematics so what I'm looking for is after choosing a basis $Phi$ is a matrix and the map can be looked as $v^* = Phi v$. The question is is there any notion or more rigorous definition, that is independent of the choice of basis that corresponds to $Phi^frac{1}{2} v$? Or is this just something that exists only in the chosen basis?
linear-algebra vector-spaces inner-product-space
linear-algebra vector-spaces inner-product-space
asked Nov 28 '18 at 2:29
Alex Botev
674314
asked Nov 28 '18 at 2:29
Alex Botev
674314
edited Nov 28 '18 at 2:38
asked Nov 28 '18 at 2:29
Alex Botev
674314
asked Nov 28 '18 at 2:29
Alex Botev
674314
asked Nov 28 '18 at 2:29
Alex Botev
674314
674314
add a comment |
add a comment |
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