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a/TFG_Sixto_José_Martínez_Gan.tex b/TFG_Sixto_José_Martínez_Gan.tex index 2685935..17d77ca 100644 --- a/TFG_Sixto_José_Martínez_Gan.tex +++ b/TFG_Sixto_José_Martínez_Gan.tex @@ -456,27 +456,26 @@ p\rar[mapsto]&\displaystyle\diffp{p}{x_i}. \dual{M} := \Hom{A}{M}{A}. \end{equation*} \end{definicion} - -{\color{red} Hago una propuesta alternativa que creo que simplifica y aclara. \begin{definicion} - Dados dos $A$-módulos $M,N$, una forma \emph{bilineal} $\Phi:M\times N\to A$ es una aplicación tal que - \begin{itemize} - \item $\Phi\!\left( a m + a'm', n \right) = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m', n \right)$ y - \item $\Phi\!\left( m, a n + a'n' \right) = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m, n' \right)$ - \end{itemize} -$\forall a,a'\in A$, $\forall m, m'\in M$, $\forall n, n'\in N$. Una forma bilineal induce homomorfismos $\Phi_1:M\to\dual{N}$, $\Phi_2:N\to\dual{M}$, dados por -$\Phi_1(m)(n):=\Phi(m, n)=:\Phi_2(n)(m)$. + Dados dos \modulo{A}s $M$ y $N$, una forma \emph{bilineal} $\Phi:M\times N\to A$ es una aplicación tal que + \begin{equation*} + \begin{cases} + \Phi\!\left( a m + a'm', n \right) = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m', n \right) \ \ \text{y} \\ + \Phi\!\left( m, a n + a'n' \right)\; = a\Phi\!\left( m, n \right) + a' \Phi\!\left( m, n' \right) + \end{cases} \quad \forall a,a'\in A, \ \forall m, m'\in M, \ \forall n, n'\in N. + \end{equation*} + Una forma bilineal induce homomorfismos $\Phi_1:M\to\dual{N}$, $\Phi_2:N\to\dual{M}$, dados por + \begin{equation*} + \Phi_1(m)(n):=\Phi(m, n)=:\Phi_2(n)(m). + \end{equation*} \end{definicion} -Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\dual{M}\to A$, definida mediante evaluación, que induce un homomorfismo natural $\Psi:=\Phi^M_1:M\to\dualdual{M}$.} - \begin{definicion} \label{Definición Módulo Reflexivo} - Sea $\psi : M \rightarrow \dualdual{M}$ el homomorfismo de \modulo{A}s que envía un elemento $m$ de $M$ a la función $\psi\!\left( m \right): \dual{M} \rightarrow A$ que a cada $\phi \in \dual{M}$ le asigna el valor $\phi\!\left( m \right)$: +Por la definición de espacio dual tenemos una forma bilineal $\Phi^M:M\times\dual{M}\to A$, definida mediante evaluación, que induce un homomorfismo natural $\Psi:=\Phi^M_1:M\to\dualdual{M}$. Esta aplicación envía un elemento $m$ de $M$ a la función $\Psi\!\left( m \right): \dual{M} \rightarrow A$ que a cada $\phi \in \dual{M}$ le asigna el valor $\phi\!\left( m \right)$: \begin{align*} - \psi : M &\longrightarrow \Hom{A}{\Hom{A}{M}{A}}{A} \\ - m &\longmapsto \psi\!\left( m \right) : \Hom{A}{M}{A} \longrightarrow A \\ - &\hspace{3.7cm} \phi \longmapsto \phi\!\left( m \right) + \Psi : M &\longrightarrow \dualdual{M} \\ + m &\longmapsto \Psi\!\left( m \right) : \dual{M} \longrightarrow A \\ + &\hspace{2.1cm} \phi \hspace{0.2cm} \longmapsto \phi\!\left( m \right) \end{align*} - Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $M$ es isomorfo a \dualdual{M} por medio de esta aplicación $\psi$. - \end{definicion} + Diremos que $M$ es un \modulo{A} \emph{reflexivo} si $\Psi$ es un isomorfismo. \begin{ejemplo} \todo{Cambiaría la estructura de este ejemplo usando bases duales y lo simplificaría.} Veamos que todo espacio vectorial de dimensión finita es reflexivo. Sea $V$ un $K$-espacio vectorial de dimensión $n$ y con base $\conjunto{v_1, v_2, \dots, v_n}$. El conjunto \Hom{K}{V}{K} se compone de las aplicaciones lineales de $V$ en $K$. Cada elemento $\phi$ de \Hom{K}{V}{K} puede representarse como la multiplicación por una matriz, puesto que