Open mapping and closed graph theorems (intro)
General · Mathematics
Study notes
Q: Why does bijective bounded T: Banach→Banach have bounded inverse? Open mapping: surjective bounded linear: OPEN map! T bijective: T⁻¹ exists; T open ⟹ T⁻¹ continuous! So T⁻¹ bounded ✓! (Bounded inverse theorem!) Closed graph: if xₙ→x, Txₙ→y ⟹ y = Tx: then T bounded!