Open mapping theorem - Imported Knowlpedia Content

Statement

Let $X$ and $Y$ be Banach spaces over the same scalar field $\mathbb R$ or $\mathbb C$ . If $T:X\to Y$ is a surjective bounded linear operator, then $T$ is an open map: $T(U)$ is open in $Y$ whenever $U$ is open in $X$ . Equivalently, there is a constant $c>0$ such that

B_Y(0,c)\subseteq T(B_X(0,1)).

By scaling and translation, this quantitative inclusion gives openness at every point. Surjectivity is essential: a proper linear subspace need not be open Conway, Chapter VI.

Proof mechanism

Because $T$ is surjective, $Y$ is the union of the closures of the sets $T(nB_X(0,1))$ . The Baire category theorem gives one such closure nonempty interior. Linearity then produces a ball about $0$ in the closure of $T(B_X(0,1))$ , and an iterative correction argument removes the closure. Completeness is used both in the category step and when the corrections are summed.

Consequences and scope

A bounded linear bijection between Banach spaces has a bounded inverse. Likewise, if $M$ is a closed subspace of $X$ , the quotient projection $X\to X/M$ is open; this identifies the quotient norm topology with the topology forced by the projection.

Completeness cannot simply be omitted. The identity from $\ell^1$ with its $\ell^1$ -norm onto the same vector space equipped with the $\ell^2$ -norm is a bounded bijection, but its target is incomplete and the inverse is unbounded. This does not conflict with the theorem because the target is not Banach.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. Springer DOI record. Relevant: Chapter VI, “Linear Operators on a Banach Space.”
Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 2, the open mapping theorem and its consequences.