Unbounded Operators 6 | Closed Graph Theorem

Closed operators on Banach spaces turn out to be automatically bounded once their domains are “large enough.” More precisely, for a linear operator T between Banach spaces X and Y, if the domain DT is a closed subset of X (equivalently, DT is itself a Banach space), then T is closed exactly when it is bounded—so continuity and closedness become the same condition in this setting. That equivalence matters because it explains why genuinely unbounded operators must have “small” domains: a closed unbounded operator cannot be defined on all of a Banach space.

The result is framed as the Closed Graph Theorem for Banach spaces. Start with a linear map T: DT → Y, where DT is assumed to be a Banach space (the transcript notes this can be arranged by restricting to DT). The operator is called closed when its graph GT = {(x, Tx) : x ∈ DT} is closed in the product space X × Y. Under the Banach-space domain assumption, closedness of the graph forces boundedness of T, and conversely boundedness forces closedness.

One direction is straightforward: if T is bounded (hence continuous), then limits pass through T. Concretely, if xn → x in X and Txn → y in Y, continuity gives Tx = y. Since x lies in the domain, the graph contains the limit point (x, y), so the graph is closed.

The harder direction uses the graph itself as a Banach space. If T is closed, then GT is a closed subspace of the Banach space X × Y, so GT is Banach. Two natural projection maps are introduced: PX: GT → X taking (x, Tx) to x, and PY: GT → Y taking (x, Tx) to Tx. The projection PX is linear and bounded, and it is also bijective because every x in X (with DT = X after restriction) appears in exactly one pair in the graph. The bounded inverse theorem (linked to the open mapping theorem) then implies that PX^{-1}: X → GT is bounded.

With PX^{-1} in hand, T can be written as a composition of bounded maps: PY ∘ PX^{-1}. Since PX^{-1} sends x to (x, Tx) and PY extracts the second component, the composition returns Tx. A composition of continuous (equivalently bounded) linear maps is continuous, so T is bounded. This completes the equivalence: for Banach-space domains, “closed graph” and “bounded operator” are the same.

An immediate takeaway for unbounded operators follows: if a linear operator is closed but unbounded, its domain cannot be a Banach space. In particular, a closed unbounded operator cannot have DT = X when X is Banach. This is why analysts spend so much effort specifying domains carefully when working with unbounded operators.