Complex Conjugate Vector Space
In mathematics, the (formal) complex conjugate of a complex vector space is the complex vector space consisting of all formal complex conjugates of elements of . That is, is a vector space whose elements are in one-to-one correspondence with the elements of :
with the following rules for addition and scalar multiplication:
Here and are vectors in, is a complex number, and denotes the complex conjugate of .
More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J (different multiplication by i).
In the case where is a linear subspace of, the formal complex conjugate is naturally isomorphic to the actual complex conjugate subspace of in .
Read more about Complex Conjugate Vector Space: Antilinear Maps, Conjugate Linear Maps, Structure of The Conjugate, Complex Conjugate of A Hilbert Space
Famous quotes containing the words complex and/or space:
“I have met charming people, lots who would be charming if they hadn’t got a complex about the British and everyone has pleasant and cheerful manners and I like most of the American voices. On the other hand I don’t believe they have any God and their hats are frightful. On balance I prefer the Arabs.”
—Freya Stark (1893–1993)
“But alas! I never could keep a promise. I do not blame myself for this weakness, because the fault must lie in my physical organization. It is likely that such a very liberal amount of space was given to the organ which enables me to make promises, that the organ which should enable me to keep them was crowded out. But I grieve not. I like no half-way things. I had rather have one faculty nobly developed than two faculties of mere ordinary capacity.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)