Nt1330 Unit 2 Math Paper

Satisfactory Essays
Suppose that A is a C-algebra and E is a linear space, which is a right A-module and the scalar multiplication satisfies (xa) = x(a) = (x)a for all x 2 E, a 2 A,  2 C. The space E is called a pre-Hilbert Amodule if there exists an A-valued map h., .i : E ! A with the following properties: (i) hx, xi  0, and hx, xi = 0 if and only if x = 0. (ii) hx, y + zi = hx, yi + hx, zi, x, y, z 2 E,  2 C. (iii) hx, yai = hx, yia, x, y 2 E and a 2 A. (iv) hx, yi = hy, xi, x, y 2 E. Such a map h., .i : E ! A is called an A-valued inner product. E is called a (right) Hilbert A-module if it is complete with respect to the norm kxk = khx, xik1/2. We note that Hilbert C-modules contain both Hilbert spaces and C-algebras. In fact, every Hilbert space
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