Logixpro Simulator Descargar Gratis Crack _TOP_ 🏴

Logixpro Simulator Descargar Gratis Crack _TOP_ 🏴

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Logixpro Simulator Descargar Gratis Crack

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Show that a linear operator $L:\mathcal{P}(E)\rightarrow \mathcal{P}(E)$ is continuous if and only if for any $F\subset E$, $L(\chi_{F})=\chi_{L(F)}$

Let $E$ be a vector space over a field $\mathbb{F}$. Let $\mathcal{P}(E)$ be the power set of $E$. Show that a linear operator $L:\mathcal{P}(E)\rightarrow \mathcal{P}(E)$ is continuous if and only if for any $F\subset E$, $L(\chi_{F})=\chi_{L(F)}$

Proof: $L$ is continuous if and only if, for any sequence $(F_{n})$ in $E$ converging to a $F$, we have $L(F_{n})\rightarrow L(F)$. Therefore, we must have $L(\chi_{F})=\chi_{L(F)}$.
I have no idea how to get the other part. Could anyone give me some help?

A:

Let $L$ be a linear transformation on $\mathcal{P}(E)$, then $L$ is continuous if and only if
$$
\forall F \subset E,\forall (a_n)_{n \in \mathbb{N}}\subset \mathbb{F} \ \ \Lambda : L\left(\chi_{F}\right)=\chi_{L(F)}\implies \Lambda(a_n\chi_{F})=a_n\Lambda\left(\chi_{F}\right)
$$
because $L\left(\chi_{F}\right)=\chi_{L(F)}$. But
$$
\Lambda(a_n\chi_{F})=\chi_{L(F)}\implies \chi_{L(F

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