Linear Algebra Done Right · Problem 8A.6

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · range-powers

Problem

Suppose T∈L(V)T \in \mathcal{L}(V). Show that V=range⁡T0⊇range⁡T1⊇⋯⊇range⁡Tk⊇range⁡Tk+1⊇⋯ .V = \operatorname{range} T^0 \supseteq \operatorname{range} T^1 \supseteq \cdots \supseteq \operatorname{range} T^k \supseteq \operatorname{range} T^{k+1} \supseteq \cdots.

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