20 1. PREREQUISITES ON OPERATORS ACTING INTO FINITE DIMENSIONAL SPACES
By the arbitrariness of z
Sm−1,
(1.3.12) S
p
sup
|z|=1
G∗z
q,
which together with (1.3.10) leads to (1.3.8).
Remark 1.5. Let 1 p ∞. Estimate (1.3.12) can be derived with the help
of the function
hz(x) =







G∗(x)z|G∗(x)z|q−2
G∗z
q/p
q
for
|G∗(x)z|
= 0,
0 for |G∗(x)z| = 0.
(1.3.13)
Indeed, since q/(q 1) = p, it follows that
hz
p
=
G∗z −q/p
q
X
|G∗(x)z|(q−1)pdμ(x)
1/p
=
G∗z −q/p
q
G∗z q/p
q
= 1.
Using (1.3.9) and (1.3.13), we obtain
S
p
= sup
f p≤1
|S(f)| (S(hz), z) =
G∗z −q/p
q
X
|G∗(x)z|qdμ(x)
=
G∗z −q/p
q
G∗z q
q
=
G∗z
q
,
which implies (1.3.12), because z is arbitrary. The case p = is treated in a
similar way.
1.4. Comments to Chapter 1
The first representation theorem in the form of Stieltjes’ integral for an ar-
bitrary linear functional in the space C[0, 1] was proved by F. Riesz [Ri1]. This
theorem was generalized to the case of functionals on C(K), where K is a compact
subset of Rn (Radon, 1913) and to the case of a compact metric space (Banach,
1937). Kakutani [Ka] proved an analogue of the Riesz representation theorem
for a compact Hausdorff space K. The first attempt to generalize this result to
noncompact topological spaces is due to Markov [Ma].
The general form of a linear functional on the space
Lp(a,
b) with 1 p
is due to F. Riesz [Ri2]. The space
L1(a,
b) was treated by Steinhaus [Ste].
Nikod´ ym [Nikd] obtained a generalization to abstract spaces with finite measure.
The material of Section 1.3 was published in [KM10].
General facts on representation of functionals in various function spaces, as well
as information of historical character can be found in the books by Aliprantis and
Border [AB], Cohn [Co], Dunford and Schwartz [DS], Edwards [Ed], Kantorovich
and Akilov [KA], Zaanen [Zaa], Maz’ya and Shaposhnikova [MSh], Section 12.5.
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