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Questions tagged [dual-maps]

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My attempts to show the dual map is isometric.

Theorem: Let $X$ and $Y$ be normed spaces such that $X\cong Y$. Let $\phi:X\rightarrow Y$ be an isometric isomorphism. Then the dual map ${\phi}^*:{Y}^*\rightarrow{X}^{*},\lambda\mapsto\lambda\circ\...
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2answers
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Proving that $\phi (T) = T^*$ is an isomorphism between vector spaces

I am tasked with the following: I am thus tasked with proving: $1)$ $\phi(T)$ is linear, so that it respects closure under scalar multiplication and addition. $2)$ $\phi(T)$ is a bijection. I only ...
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0answers
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Any connection between the adjoint map that has determinant $det(\phi)^{(n-1)}$, and the adjoint map that has determinant $ det\phi$?

Is there any connection between the adjoint mapping that is introduced while studying the matrices, and the adjoint mapping that is introduced while studying inner product spaces ? I mean, for ...
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A question about the dual of super vector space

Let $V$ be a vector space over a field $K$. Denote the dual of $V$ by $V^{*}$, that is $V^*=Hom_K(V,K)$. Suppose there is a morphism $\alpha: V \rightarrow V$. Then we know $\alpha$ induces a morphism ...
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Bilinear Maps and their relationships with dual bases

I have a theorem without proof. I have searched many books and tried on myself, but i still dont have the solution. Let M and N F-vector spaces, T be a base of M ,S be a base of N such that dimension ...