Unbounded Operators 1 | Introduction and Definitions

Unbounded operators sit at the center of functional analysis because they are unavoidable in the mathematics of partial differential equations and quantum mechanics—yet they cannot be handled using the usual “bounded operator” framework. In quantum theory, position and momentum are represented by operators whose order matters: the composition XP differs from PX, with their difference tied to the identity operator (up to a factor involving the imaginary unit). That commutation behavior cannot be reproduced if only bounded operators are allowed, forcing a broader notion of linear operators—one that can be unbounded.

The course begins by setting up the basic objects needed to define unbounded operators rigorously. Two normed vector spaces, X and Y, are fixed over a common field F, which is either the real numbers or the complex numbers (with complex spaces suggested as the default mental model). Unlike bounded operators, an unbounded operator does not necessarily act on all of X. Instead, it is defined on a subspace D ⊂ X called the domain, and maps elements of D into Y via a linear rule T: D → Y. This “domain plus action” viewpoint matters because many notational conventions in the literature compress these two ingredients—sometimes writing T ⊂ X → Y, sometimes writing T with an explicit domain, and sometimes using shorthand like D(T).

A key class for the upcoming theory is “densely defined” operators. The operator T is densely defined when its domain D is dense in X, meaning the closure of D equals the whole space X. This density requirement is only genuinely interesting in infinite-dimensional spaces: in finite dimensions, a proper subspace cannot be dense, so the distinction collapses.

To prepare for later results, the course recalls standard operator subspaces—range and kernel—while explicitly accounting for the domain. The range of T consists of all outputs T(x) with x in D, forming a subspace of Y. The kernel consists of all x in D such that T(x) = 0, forming a subspace of X.

The transition to boundedness is then made precise. A linear operator T is bounded if there exists a finite constant C > 0 such that for every x in D, the norm of T(x) in Y is at most C times the norm of x in X. In that case, the operator norm stays finite. If no such finite C exists—equivalently, if the operator norm becomes infinite—then T is called unbounded. The course also emphasizes an important equivalence from functional analysis: for linear maps, boundedness is equivalent to continuity at all points in the domain. Therefore, unbounded operators are necessarily discontinuous everywhere on their domain.

The practical takeaway is clear: unbounded operators are not a pathological corner case. They are required by the structure of quantum mechanics and appear naturally in PDEs, and their defining feature is that they act on dense domains while failing to be continuous—something that can only happen in infinite-dimensional settings. The next step is to move from definitions to concrete examples.