let H be a Hilbert space a, b belongs to H -{0} are orthogonal element and U : H ----> H is defined by U(x) = a < x, b > + b < x, a> . calculate || U ||
Answers
Answer:
I hope it will help you
Step-by-step explanation:
1. The Hilbert space L2 157
The resulting L
2
(R
d
)-norm of f is defined by
kfkL2(Rd) =
µZ
Rd
|f(x)|
2
dx¶1/2
.
The reader should compare those definitions with these for the space
L
1
(R
d
) of integrable functions and its norm that were described in Sec-
tion 2, Chapter 2. A crucial difference is that L
2 has an inner product,
which L
1 does not. Some relative inclusion relations between those spaces
are taken up in Exercise 5.
The space L
2
(R
d
) is naturally equipped with the following inner prod-
uct:
(f, g) = Z
Rd
f(x)g(x) dx, whenever f, g ∈ L
2
(R
d
),
which is intimately related to the L
2
-norm since
(f, f)
1/2 = kfkL2(Rd)
.
As in the case of integrable functions, the condition kfkL2(Rd) = 0 only
implies f(x) = 0 almost everywhere. Therefore, we in fact identify func-
tions that are equal almost everywhere, and define L
2
(R
d
) as the space
of equivalence classes under this identification. However, in practice it is
often convenient to think of elements in L
2
(R
d
) as functions, and not as
equivalence classes of functions.
For the definition of the inner product (f, g) to be meaningful we need
to know that fg is integrable on R
d whenever f and g belong to L
2
(R
d
).
This and other basic properties of the space of square integrable functions
are gathered in the next proposition.
In the rest of this chapter we shall denote the L
2
-norm by k · k (drop-
ping the subscript L
2
(R
d
)) unless stated otherwise