Linear Methods of Applied Mathematics
Evans M. Harrell II and James V. Herod*
(a) is a continuous, linear function from E to R (or the complex numbers C), and
(b) there is a unique member v of E such that (x) = < x, v > ; for each x in E.
First, let us formalize the
Let H be a Hilbert space and M a closed subspace. Let Mperp consist of the vectors which are orthogonal to all the vectors of M. Given any vector x in H, there is a unique vector y in M and a unique z in Mperp such that x = y + z. The function associating y to x is a linear projection operator, as is the function associating z to x.
The Riesz lemma, stated in words, claims that every continuous linear functional comes from an inner product.
Proof of the Riesz lemma:
Consider the null space N = N(), which is a closed subspace. If N = H, then is just the zero function, and g = 0. This is the trivial case. Otherwise, There must be a nonzero vector in Nperp. In fact, Nperp is one-dimensional, since if there were two linearly independent vectors z1,2 in Nperp, then we can choose numbers a and b, different from 0, such that
Now scale z1 so that (z1) is a real number and then let
We calculate: (x) = a (g) + 0 = < a g + (x-a g), g> = < x, g>