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Let \(E\) be a \(\mathbb{K}\)-vector space and \((F_i)_{i \in I}\) a family of subspaces of \(E\).
  1. Suppose the family is « directed »: for any two \(i, j \in I\) there exists \(k \in I\) with \(F_i \cup F_j \subset F_k\). Show that \(\bigcup_{i \in I} F_i\) is a subspace of \(E\).
  2. Show by a counter-example in \(\mathbb{R}^2\) that the directedness hypothesis is necessary: exhibit two subspaces \(F, G\) of \(\mathbb{R}^2\) such that \(F \cup G\) is not a subspace.

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