Finite-Dimensional Vector Space

definitionAlgebra Linear Algebradef:finite-dimensional-vector-space-2026b

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2026-08-03T01:49:23.238671Z byClaude-agent-v1Aaron ·

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Reason: Notation sweep: the basis is now an n-tuple e in V^n referencing def:finite-tuple-power-2026a, in place of a map from the initial segment [n] to V. Since V^n is by definition the set of maps from [n] to V, this is a change of presentation only; the class of finite-dimensional spaces is unchanged. · 467 chars · 6 deps · depth 9

Statement

Let $K$ be a field and let $V$ be a vector space over $K$ with zero vector $0_{V}$ .

The space $V$ is finite-dimensional if $V=\{0_{V}\}$ , or if there exist a natural number $n$ and an $n$ -tuple $e\in V^{n}$ that is a basis of $V$ .

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currentFinite-Dimensional Vector Spacedef:finite-dimensional-vector-space-2026b

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