A linear operator on a Hilbert space is densely defined when its operator domain is a dense subset of . Density makes the adjoint operator unique wherever its defining inner-product identity has a solution.
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In functional analysis, a densely defined operator is a linear operator defined on a dense subset of a vector space (usually a Hilbert space or a Banach space). Specifically, if \( A \) is an operator acting on a vector space \( V \), we say that \( A \) is densely defined if its domain \( \mathcal{D}(A) \) is a dense subset of \( V \).