bijective bounded linear operators have bounded linear inverses

[[concept]]

Corollary

If $B_{1}, B_{2}$ are Banach spaces and $T : B_{1} \to B_{2}$ is a bijective bounded linear operator, then $T^{- 1} \in B (B_{2}, B_{1})$ is also.

Proof

$T^{- 1}$ is continuous $⟺$ for all $U \subset B_{1}$ open, $(T^{- 1})^{- 1} (U) = T (U)$ is open. And this is true by the open mapping theorem

Mentions

File	Last Modified
Functional Analysis Lecture 4	2025-10-02

const { dateTime } = await cJS()

return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}