finite vector space

[[concept]]

finite vector space

$V$ is finite if every linearly independent set is finite.

ie, if for every set $E \subseteq V$ such that $\forall v_{1}, \dots, v_{N} \in E$

\sum_{i = 1}^{N} a_{i} v_{i} = 0 ⟹ a_{1} = \dots = a_{N} = 0

and $E$ has finite cardinality, then $V$ is finite-dimensional. Otherwise, $V$ is infinite-dimensional.

Example

$C ([0, 1])$ is infinite dimensional because

E = {f_{n} (x) = x^{n} | n \in N \cup {0}}

is a linearly independent set with non-finite cardinality.

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Functional Analysis Lecture 1	2025-09-02
Functional Analysis Lecture 19	2025-08-18

Created 2025-05-27 ֍ Last Modified 2025-09-11