Real Analysis with Economic Applications - Chapter A pot
Real Analysis with Economic Applications - Chapter A pot
... pretty ric h algebraic
structure. In particular, it is a partially ordered linear space (see Chapter F).
Wh en the dom ain of a real function is a poset, we can talk about how this map
a ects the ... exists a function
f : A →
V
A such that f (A) ∈ A for each A ∈ A.
One can reword this in a few other wa y s.
12
But, how about the following algorithm? Start with A
1...
Real Analysis with Economic Applications - Chapter E potx
... an arbitrary function, i =1, , m. This
sort of a game i s called an aggregative game.
28
As we shall see below, man y in ter-
esting economic games are aggregative. (See Corchó n (1996) for a ... ,m
} in game theory; we also adopt this no-
tation below.) We say that a strategic game G is finite i f e ach action space X
i
is
finite.
By far t he most f amous example o f a s trategic gam...
Real Analysis with Economic Applications - Chapter F pot
... A real- valued linear operator is r eferred to as a linear fun ct ion a l on X.
17
Notation. T h e s et of all linea r o perators from a linear sp ace X in to a line ar space
Y is denoted as ... ∈ N and fixsomeA ∈ 2
N
\{∅}. The capacity u
A
∈ V
N
defined as
u
A
(B):=
1 ,A B
0 ,A- B
(15)
is called a unan im ity gam e . We can again think of this game as corresponding to
a vo...
Real Analysis with Economic Applications - Chapter H pot
... arbitrage a ssociated with a p rice functional π is a state-
contingent claim a ∈ R
Ω
such tha t a | 0 and π (a) ≤ 0, or a 0 and π (a) < 0. Thus,
w e say that the price functional π is arbitrage-free ... contains a d ollars and a state-contingent claim a. By a
standard agen t f or Ω in t he presen t s etting we me an a concav e function u : Z
Ω
→ R
such that
b> ;...
Real Analysis with Economic Applications - Chapter B ppt
... e ntable in
view of Ex ercise 24. Tak e any real a, an d p ick any z
a
∈ Y such that (a, 1)
lex
z
a
and z
a
lex
(a, 0). Clearly, we mu st have z
a
1
= a and z
a
2
∈ (0, 1). But then z
a
= ... Then, for any a ∈ R,wehave
u (a, a +1)>u (a, a) so that I (a) :=(u (a, a) ,u (a, a +1))is a nondegenerate interval
in R.Moreover,I (a) ∩ I(b)=∅ for any a = b, for w...
Real Analysis with Economic Applications - Chapter C pps
... speak of
p
as a metric space, we always have this metric in m in d !) O f
course,wehavetocheckthatd
p
is well-defined as a real- valued function, and that
it satisfies the triang le inequ ality. ... multidimensional
analogue of the bisection argum ent given above. In Section 4 w e will give an alter-
native proo f.
Exercise 26. Let n ∈ N, and take any real numbers a and b with b>...
Real Analysis with Economic Applications - Chapter D ppsx
... For any real numbers a and b with a& lt;b,show that C[0, 1] and
C [a, b] are isometric.
Exercise 18. Let
d and D betwometricsonanonemptysetX.
(a) Show that d and D are equivalent iff id
X
is an homeomorphism.
(b)(Carothers)
(X, ... viewed as a functional equation in w hich the unkno w n variable
is a function — it is often ca lled Cauchy’s (first) fun c tio n al equat io n.
25
In ge...
Real Analysis with Economic Applications - Chapter G pptx
... i
functionals? T he main reason is that these functionals g iv e us a n analytic charac-
terization of an arbitrary algebraically closed convex set with nonempt y a lgebraic
346
[4] The linear operator D : ... id es a novel application of Proposition 4 to linear algebra.
E{dpsoh 13. (Farkas’ L emma)Takeanym ∈ N, and any u
1
, , u
m
,v ∈ R
n
. A
famou s result of linear algebra , Far...
Real Analysis with Economic Applications - Chapter I ppt
... space X. Translation in variance is thus a concept whic h is partly met-
ric a nd partly algebraic — it connects in a particular w ay the distance function o n a
linear space with the operation of ... functional analysis and operator
theory in a substantial way — his contributions have accen tuated the development of quantum
mechanics early in the 20th century. There are at leas...
Real Analysis with Economic Applications - Chapter J doc
... not have a Schauder basis;
(e)If
X has a countable Schauder basis, then it i s separable.
Remark 1. One major reason wh y an n-dim e ns ional linea r space is a well-behaved
entit y is th at there ... think of an infinite dimensional Banach
space as a “large” n orm ed linear space. In particular, an y
p
is, intuitively, “much
larger” than any of its subspaces that are spanned by coun...
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