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OffHakhol
  • Member for 3 months
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About

Self learner sometimes doing exercice without correction. I hope to get help here.
English is not my mother tongue. Sorry in advance for any mistakes.


A lot of my questions on algebra can be founded here. https://webusers.imj-prg.fr/~antoine.ducros/


Soit $(X;A;\mu)$ un espace mesure et $f$ une fonction mesurable sur $X$ ( a valeurs dans $ \mathbb{R} \cup +\infty$ ou $ \mathbb{C} $ ). L'expression $ \int f d \mu $ a un sens dans l'un des cas suivants:
1- $f$ est une fonction a valeurs reelles positives (eventuellement meme on peut avoir $\int f d \mu = + \infty$ )
2- Soit $ \int |f| d \mu < \infty $ cad $f$ est dite $(X, \mu)-$ integrable. Mais aussi dans un tel cas on a $\int f d \mu$ est un nombre dans $ \mathbb{C} $ ou $\mathbb{R}$.


Question

Let a linear mappimg $\phi : E \rightarrow F$ between two normed vector space $(E; \| . \|_E)$ and $(F; \| . \|_F)$.
a) Prove that in $ \mathbb{R}^+ \cup \infty $. We have:
$$ \| \phi \| =_{def} Sup \{ \frac{ || \phi(x) ||_E}{ \| x \|_F} : x \in E - \{ 0\} \} = Sup \{ || \phi(x) ||_F : \| x \|_E \leq 1 \} = Sup \{ || \phi(x) ||_F : \| x \|_E = 1 \} = min \{ C \in \mathbb{R} \; verifying \; \forall x \in E , \| \phi(x) \|_F \leq C \| x \|_E \}$$
b) And that $ \phi $ is continuous on iff $ \| \phi \| < \infty$

Answer

a)

b)

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