Definition For Bounded Operators
Suppose H is a Hilbert space, with inner product . Consider a continuous linear operator A : H → H (this is the same as a bounded operator).
Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : H → H with the following property:
This operator A* is the adjoint of A. This can be seen as a generalization of the adjoint matrix of a square matrix which has a similar property involving the standard complex inner product.
Read more about this topic: Hermitian Adjoint
Famous quotes containing the words definition and/or bounded:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Me, what’s that after all? An arbitrary limitation of being bounded by the people before and after and on either side. Where they leave off, I begin, and vice versa.”
—Russell Hoban (b. 1925)