By Ramon E. Moore

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We will now show Linear functionals 7] 39 that hf = f(h)h satisfies the conclusion of the theorem. To see this, note that f(x)h -f(h)x is in N(f) for all x in H, because f(f(x)h -f(h)x) =f(x)f(h) -f(h)f(x) = 0Thus, (h, f(x)h -f(h)x) = 0, so f(x) = (h, f(h)x). It follows that f(x) = (x, f(h)h) for all x in H. For uniqueness, if (x, h) = (x, q) for all x in H, then in particular, for x = h - q, we have (h - q, h - q) = 0; so h = q. This completes the proof. Definition If f is a bounded linear functional on a normed linear space, then sup x*o If(x)I l Ilxll _ sup If(x)I IIx11=1 is called the norm off and is denoted by 11f ll.

Examples (1) For X = Y = E" (which can be viewed as the linear space of real valued functions on the first n positive integers), a linear transformation, represented by an n-by-n matrix, is a linear operator. (2) For X = C[0, 11 and Y = R, a linear functional is a linear operator, from X into Y. (3) For X = Y = C [0, 11 and K continuous on [0, 1 ] X [0, 1 ] , L J) (t) = f K(t, t') f(t') dt' 1 0 defines a linear integral operator, mapping X into itself. (4) For X = C2 [0, 1 ] , twice continuously differentiable real valued functions on [0, 1 ],and Y = C[0, 1 ] , L(f) (t) = a(t) f "(t) + b(t)f'(t) + c(t) f(t) defines a linear differential operator L from X into Y, if a, b and c are continuous.

C,. ,n, for some point x*, then x -x* is orthogonal to every point in M because of the linearity of the inner product with respect to the second argument. To see whether there is such a point x* in M, put x* = c;x1 + c2x2 + ... + enxn Again, using the linearity of the inner product (this time with respect to the first argument), the above system of equations can be put into the matrix form (XI, X0 (X2, X1) ... (xn,x1) CI (X, X1) (x1,x2) (x2,x2) ... (xn,x2) c2 (x, x2) Cn \(x,xn) \(xi,xn) (X2, Xn) ...