grandes-ecoles 2023 QII.2

grandes-ecoles · France · x-ens-maths-c__mp Proof Proof of Equivalence or Logical Relationship Between Conditions

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.

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