Remark 1
Tag 0797, permanent. Cite it as [Bourbaki, Tag 0797].
Topological Vector Spaces, I, p. 12, § 2. Linear varieties in a topological vector space
The family of topologically independent sets of a topological vector space is not necessarily inductive for the relation of inclusion (I, p. 25, exerc. 2); this situation is thus different to that for algebraically independent sets. Moreover there does not necessarily exist in E a maximal topologically independent subset (I, p. 25, exerc. 4), thus there does not necessarily exist a subset that is both total and topologically independent.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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