Prove the representaion theorem in Hilbert spaces H by finding a point in a plan

Question

Prove the representaion theorem in Hilbert spaces H by finding a point in a plane with the minimum distance from the origion

Explanation / Answer

if T is obviously continuous at x = 0 and hence by Exercise 3.1 it is continuous on X. Conversely, suppose that T is continuous on X, and is hence continuous at x = 0. There is then d > 0 such that if kxk = d, then kT xk = 1. Let now x ? X and x 6= 0, then k d kxkxk = d, so kT ( d kxkx)k = 1. Thus kT xk = 1 d kxk. If we choose C = 1 d , then (3.1) holds for x 6= 0. But when x = 0, (3.1) holds always. ¿From this theorem it follows that if T is a continuous linear transformation from X into Y , then kT k := sup x?X , x6=0 kT xk kxk < +8 (4.2) and is the smallest C for which (3.1) holds. kT k is called the norm of T . Of course, kT k can be defined for any linear transformation T from X into Y , and T is continuous if and only if kT k < +8. Hence a continuous linear transformation is also called a bounded linear transformation.

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