Linear operators and their adjoints

Abstract

Let H ₁, and H ₂ be vector spaces over $\mathbb{K}$. A linear operator T from H ₁ into H ₂ is, by definition, a linear mapping of a subspace D(T) of H ₁ into H ₂. The subspace D(T) is called the domain of T. The image R(T)= T(D(T)) = {Tf: f ∈ D(T)} is called the range of T. Since we only treat linear operators here, we shall speak only about operators from H ₁, into H ₂. If H ₁, = H ₂ = H, then T is called an operator on H. A linear operator from H into $\mathbb{K}$ is called a linear functional. The range of an operator T from H ₁, into H ₂ is a subspace of H ₂. An operator is injective if and only if Tf=0 implies f = 0.