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Unformatted text preview: MATH 131B
1ST PRACTICE MIDTERM Problem 1. State the book’s deﬁnition of:
(a) Uniform convergence of a sequence of function
(b) A metric
(c) Pointwise convergence of a sequence of functions
(d) A continuously differentiable function of two variables
(e) A continuous function of two variables
(f) The supremum norm
¤ ¡
¥£¢ Problem 2. Prove that a uniform limit of a sequence of continuous functions is also continuous.
©
¨ ¦
§¡ be a sequence of functions that converges uniformly on ¡ to a function .
4 # '% #!
5)3&$2¡ Problem 3. Let
Show that 0 # '% #! ¦
1)(&$"§¡ Give an example which shows that the conclusion does not hold if we only assume pointwise
convergence.
be the space of functions on A9
¥B@¨ Problem 4. Let
Let which have a continuous derivative. A 9¨ 7
¥B@¢86 4 ¤ Q ¡ H ¤ ¡ D ¡
TGSR¢PIGC¢FEC¢ is a norm
is complete in this norm.
aD
b# D
C% if
and
if
.
is Cauchy for this metric if and only
whenever
. (c) Show that is
9 `
X ` A W e
sr , but does not converge
%#
&Y! Q ¡ A9
¥B@¨ 9 X e v B¦
w
wv
xB¦ D¦
ut¡
y
7 be a continuously differentiable function. Suppose that
is a “uniform contraction”: there is a constant
.
q
2p on W W converges pointwise to A9
6
A ¡9
¢B@¨
5` 1 ` A9
¥BVU¨ 7 6 h# D ¦
igfR#
c
d
` ¦&# X X
#
Y!
#
6
1
%
` 0
¡ Problem 7. Let
for all
. Show that
for all
, #
Y! Problem 6. Show that
uniformly. D
£% Problem 5. Endow with the following metric :
(a) Show that is a metric. (b) Show that a sequence
if it is eventually constant; i.e., for some ,
complete with respect to this metric. D
1# ¡
C¢ (a) Show that
(b) Show that so that A9
¢B@¨ ! ¡ % #!
b2&Y¡
¥¥@U¨
A9
A9
¥¥@U¨ c
# c `
# A ...
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This note was uploaded on 09/19/2011 for the course MATH 131B taught by Professor Hitrik during the Winter '08 term at UCLA.
 Winter '08
 hitrik
 Math

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