Unbounded Operators 5 | Example

A concrete example in ℓ² shows how easy it is to build an unbounded linear operator that fails even the basic requirement of closability. The setup uses the Hilbert space X = ℓ² (square-summable complex sequences) and a dense domain consisting of all finite linear combinations of the canonical unit vectors E₁, E₂, E₃, … . The operator T maps each unit vector Eⱼ to the complex number j, and then extends linearly to the whole span of these unit vectors.

The first key result is that T is unbounded. Computing the operator norm means taking the supremum of ||Tx|| over all inputs x with ||x|| = 1. Restricting attention to unit vectors already forces the supremum to blow up: ||T(Eⱼ)|| equals |j| (since the codomain is ℂ with its usual absolute value norm). Because |j| grows without bound as j increases, no finite operator norm exists. So T is unbounded, as expected for many natural “coordinate-like” maps on infinite-dimensional spaces.

The second key result is more striking: T is not closable. Closability can be tested using a sequence criterion: if xₙ in the domain tends to 0 and Txₙ tends to some y, then closability forces y = 0. Here, T is deliberately arranged so that images can converge to a nonzero limit even when the inputs collapse to 0.

To produce the contradiction, one starts with a sequence in the domain that converges to 0 but whose images do not converge to 0. The argument then refines this by selecting a subsequence whose images stay a fixed distance away from 0—so |Txₖ| ≥ ε for some ε > 0. With that safety margin, a new sequence zₖ is defined by normalizing: zₖ = xₖ / (Txₖ). This keeps zₖ inside the domain and ensures zₖ → 0 in ℓ², because dividing by the nonvanishing scalars does not prevent convergence to 0.

Yet the images behave differently: Tzₖ = T(xₖ)/(Txₖ) = 1 for every k. Therefore Txₖ (after normalization) converges to y = 1 in ℂ, while the corresponding inputs converge to 0. The closability criterion would require y = 0, so the nonzero limit produces a direct contradiction. The conclusion is that T is not closable—and consequently it is also not a closed operator.

The broader takeaway is practical: in infinite-dimensional settings, unbounded operators can fail closability with little effort, even when they look simple on a dense domain. This example serves as a warning while setting up later “positive” cases where unbounded operators are closed or closable under additional structure.