Let \(X\) and \(Y\) be normed spaces. The statements are on Appendix: Functional analysis.
| Hypothesis | Theorem | Conclusion |
|---|---|---|
| \(X\) Banach; \(T_\alpha\colon X\to Y\) bounded linear with \(\sup_\alpha\norm{T_\alpha x}<\infty\) for each \(x\) | uniform boundedness | \(\sup_\alpha\norm{T_\alpha}<\infty\) |
| \(X, Y\) Banach; \(T\colon X\to Y\) bounded linear and surjective | open mapping | \(T\) is open; if \(T\) is also injective, \(T^{-1}\) is bounded |
| \(X, Y\) Banach; \(T\colon X\to Y\) linear with closed graph | closed graph | \(T\) is bounded |
| \(M\subseteq X\) a subspace; \(f\) a bounded linear functional on \(M\) | Hahn–Banach | an extension \(F\in X^*\) of \(f\) with \(\norm F = \norm f\) |
The uniform boundedness principle applies the Baire category theorem to the complete space \(X\). The open mapping theorem applies the Baire category theorem to \(Y\) and uses the completeness of \(X\) to sum an absolutely convergent series. The closed graph theorem follows from the bounded inverse theorem applied to the projection from the graph of \(T\), a Banach space, onto \(X\). The Hahn–Banach theorem is proved using Zorn’s lemma and holds for every normed space \(X\).
Consequences
If \(X\) is a Banach space and \(T_n\colon X\to Y\) are bounded linear maps such that \(Tx\coloneqq\lim_n T_nx\) exists for every \(x\in X\), then \(T\) is a bounded linear map.
\(T\) is linear as a pointwise limit of linear maps. Each sequence \((T_nx)_n\) converges, hence is bounded, so the uniform boundedness principle gives \(C\coloneqq\sup_n\norm{T_n}<\infty\), and \(\norm{Tx} = \lim_n\norm{T_nx}\leq C\norm x\).
To show that the graph of a linear map \(T\colon X\to Y\) is closed, it suffices to show that \(x_n\to x\) and \(Tx_n\to y\) imply \(y = Tx\).
If \(X\) is a normed space and \(x\neq y\) in \(X\), there is \(f\in X^*\) with \(f(x)\neq f(y)\). In particular, the weak topology on \(X\) is Hausdorff.
By the Hahn–Banach theorem there is \(f\in X^*\) with \(\norm f = 1\) and \(f(x-y) = \norm{x-y}\neq0\).
Hilbert spaces
In a Hilbert space \(H\), every closed subspace \(M\) has an orthogonal projection \(P\colon H\to M\) with \(H = M\oplus M^\perp\), the Riesz representation theorem identifies \(H^*\) with \(H\), and for an orthonormal basis \(\theset{e_n}\) Parseval’s identity gives \(\norm x^2 = \sum_n\abs{\inner x{e_n}}^2\). In particular, for a bounded linear functional \(f\) on a closed subspace \(M\), \(f\circ P\) extends \(f\) to \(H\) with \(\norm{f\circ P}=\norm f\). See Banach and Hilbert spaces.