00:01
In this question we have been given that t is a bounded linear self -ad joint operator on a helbert space h.
00:08
We need to show that norm of t is equal to supremum of mod of inner product of x and dx.
00:16
Okay, so let us start with this.
00:20
So let x be non -zero.
00:24
Then consider consider this inner product of t x and x so it will be equal to norm of x square mod of inner product of t norm x t norm x inverse x comma norm of x in a product than modernist so this is obviously going to be less than or equal to supreme of inner product of t u and u times norm of x square where norm of u is just one so let's say this is our equation number one correct now one thing we know okay so since we know that four times of real part of the inner product dxy is equal to, it is equal to inner product of t of x plus y, comma, x plus y, correct, minus inner product of t of x minus y, comma, x minus y, correct.
01:50
Now we have, so this implies that four times of real part of tx, y, is equal to, sorry, it is less than or equal to, less than or equal to the supreme of inner product of t u which u, where norm of u is equal to 1 times norm of x plus y, norm of x plus y, square, plus norm of x minus y square, correct? so this can be written it to be equal to supremum when norm of u is equal to one, inner product of t u with you, two times of norm of x square plus norm of y square...