grandes-ecoles 2023 QII.4

grandes-ecoles · France · x-ens-maths-c__mp Proof Deduction or Consequence from Prior Results

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. We seek to show the following theorem:
Theorem 1: The following two properties are equivalent: - (P1) $A$ is relatively compact. - (P2) $A$ is equicontinuous and for all $x \in K$, the set $A(x) = \{f(x) \mid f \in A\}$ is bounded.
Show that $(P1) \Rightarrow (P2)$.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. We seek to show the following theorem:

Theorem 1: The following two properties are equivalent:
- (P1) $A$ is relatively compact.
- (P2) $A$ is equicontinuous and for all $x \in K$, the set $A(x) = \{f(x) \mid f \in A\}$ is bounded.

Show that $(P1) \Rightarrow (P2)$.

Paper Questions

QI.1 QI.2 QI.3 QII.1 QII.2 QII.3 QII.4 QII.5 QII.6 QII.7 QIII.1 QIII.2 QIII.3 QIII.4 QIII.5 QIII.6 QIII.7 QIV.1 QIV.2 QIV.3