OffHakhol
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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 $$\| \phi \| < \infty$$

a)

b)

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