ABSTRACT
This chapter discusses the class of linear transformations from the vector space V to the vector space W by L. Since the scalar field F is a vector space, one class of linear transformations from V is L. The topological dual consists of continuous linear functional. The adjoint of a linear operator has classically been defined for all vector spaces, but the most common applications occur with inner product spaces. It has become common in the literature to both differentiate between the adjugate of a linear transformation on a vector space and reserve the term adjoint for inner product spaces, or to only deal with inner product spaces. In some areas of science, including mathematical physics, eigenvectors on inner product spaces are of central importance. The spectral theorem is arguably the most important theorem in linear algebra. The proof of the spectral theorem in the real case is more difficult than in the complex case.
