Unbounded Operators 1 | Introduction and Definitions [dark version]

Unbounded operators are essential in functional analysis because they naturally arise in both partial differential equations and quantum mechanics—settings where the usual “bounded operator” framework breaks down. In quantum mechanics, the position and momentum observables are represented by operators X and P, and the non-commutativity of measurements shows up as a specific operator relation: the compositions XP and PX differ by a term proportional to the identity (with an imaginary unit involved). That kind of relation cannot be satisfied if one restricts attention to bounded operators alone, forcing a broader definition of linear operators.

The course then sets up the mathematical groundwork needed to talk about operators that may be unbounded. Start with two normed vector spaces, X and Y, over a common field F (either real numbers or complex numbers, with complex spaces suggested as the default in most applications). Unlike bounded operators, an unbounded operator is not required to act on the entire space X. Instead, it is defined only on a subspace D ⊆ X, called the domain. A linear operator T maps each x in D to an element T x in Y.

Because different texts use different shorthand, the transcript emphasizes notation: some authors write T ⊆ or omit the explicit domain in the symbol, while others treat an operator as the pair of its action together with its restriction to D. For this course, the key new concept is “densely defined” operators. An operator T is densely defined when its domain D is dense in X, meaning the closure of D equals all of X. This matters only in infinite-dimensional spaces: in finite dimensions, any proper subspace cannot be dense, so the phenomenon of having a smaller but dense domain is essentially an infinite-dimensional feature.

With the domain in place, standard operator substructures must be defined with care. The range (or image) of T consists of all outputs T x with x drawn from D, producing a subspace of Y. The kernel consists of all x in D such that T x = 0, forming a subspace of X. These definitions mirror the bounded case but explicitly respect the restricted domain.

The transcript then recalls boundedness. A linear operator T is bounded if there exists a finite constant C 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. Equivalently, the operator norm stays finite. If no such finite C exists—so the operator norm effectively becomes infinite—then T is called unbounded. A crucial equivalence from functional analysis ties boundedness to continuity: a linear map is bounded if and only if it is continuous at all points of its domain. Therefore, unbounded operators must be discontinuous everywhere on their domain. The course flags this as another reason unbounded operators are inherently tied to infinite-dimensional settings, where continuity can fail in ways impossible in finite dimensions.

Overall, the definitions build a bridge from physical and PDE motivations to a precise functional-analytic framework: linear operators with restricted, often dense domains, whose failure of boundedness corresponds to pervasive discontinuity—exactly the behavior needed to model quantum observables and differential operators.