Monday:
- Work:
- We want to show that the metrics induced on $\mathbf {R}^n$ by
the $p$-norms are all strongly equivalent for $1\le p \le\infty$.
- Observe that it will suffice to show that given $1\le p\le\infty$
there are $c,d>0$ such that $$c\|\mathbf x\|_2\le \|\mathbf x\|_p
\le d\|\mathbf x\|_2\tag{*}$$ for all $\mathbf x\in\mathbf{R}^n$.
- Prove (*) using the fact that a continuous function on a
closed bounded subset $C$ of $\mathbf{R}^n$ attains its maximum
and minimum on $C$. (Why is "attains" important above?)
- Give an example of a metric on $\mathbf{R}^n$ which is not
equivalent to any of the metics in problem 1.
- $L^p$-spaces: Since the Math 103 syllabus usually does
not cover $L^p$-spaces, I thought I would include
some optional exercises to at least see that $\|\cdot\|_p$
is a semi-norm for $1\le p \le \infty$. These do not have to be
turned in and if you're that lazy, you can find all this and more
in Folland's Real Analysis Section 6.1. Throughout,
$(X,{\mathcal M},\mu)$ is meant to be a measure space and $f$ and
$g$ measurable functions on $X$. If you are truely measure
adverse, we can replace $\mathcal L^p(X)=\{f:X\to \mathbf C: \int_X
|f(x)|^p\,d\mu(x)<\infty\}$ with $C([0,1])$ and show that
$\|\cdot\|_p$ is a norm on $C([0,1])$.
- OP: Show that if $a,b\ge0$
and $0<\lambda<1$, then $a^\lambda b^{1-\lambda} \le \lambda a
+ (1-\lambda) b$. (Hint: reduce to the case $b>0$. Then let
$t=a/b$ and use calculus to prove that $t^\lambda -\lambda t\le
1-\lambda$.)
- OP: (Holder's
Inequality): Suppose $1< p < \infty$ and
$\frac 1p+\frac 1q=1$. (We call $q$ the conjugate
exponent to $p$.) Show that $\|fg\|_1\le\|f\|_p\|g\|_q$.
I suggest the following:
- Reduce to the case $\|f\|_p=1=\|g\|_q$. (Use the fact
that the norms are homogeneous and beware of norms equalling $0$ and
$\infty$.)
- Use 3 to show that $|f(x)g(x)|\le \frac 1p |f(x)|^p + \frac 1q
|g(x)|^q$. (Let $a=|f(x)|^p$, $b=|g(x)|^q$, and $\lambda=\frac 1p$.)
- Integrate.
- OP:
(Minkowski's Inequality): Let $1\le p \le
\infty$. Show that $\|f+g\|_\infty\le \|f\|_p+\|g\|_p $
provided both $f$ and $g$, and hence $f+g$, are in $\mathcal
L^p$. I
suggest the following:
- Show that if $\|f\|_\infty<\infty$, then $\{\, x:
|f(x)|>\|f\|_\infty\,\}$ is $\mu$-null.
- Reduce to the case $1< p < \infty$.
- Observe that if $q$ is the conjugate exponent to $p$,
then $(p-1)q=p$ and apply Holder to $|f+g|^p\le
(|f|+|g|)|f+g|^{p-1}$.
|
Wednesday:
Work: Keep in mind that problems marked with
" OP: " are optional and not to
be turned in.
- OP:
Let $E$ be a subset of a metric space $X$. We say that $x$
is a limit point of $E$ if there is a sequence
$(x_n)\subset E$ such that $x=\lim_{n\to\infty} x_n$. Show that
$E$ is closed if and only if $E$ contains all its limit points.
- OP:
State and prove a result characterizing open sets in a metric
space interms of sequences (as we did for closed sets in the
previous problem). The following terminology might be useful.
If $E$ is a subset of a metric space $X$, then a sequence
$(x_n)\subset X$ is eventually in $E$ if there is a $N$
such that $n\ge N$ implies $x_n\in E$.
- Let $(X,\rho)$ be a metric space. If $A\subset X$, then
define $\rho(x,A)=\inf\{\, \rho(x,y):y\in A\,\}$.
- Show that $\rho(x,A)=0$ if and only if $x\in \overline A$.
- Show that $x\mapsto \rho(x,A)$ is continuous.
- Show that if $A$ and $B$ are disjoint nonempty closed
subsets of $X$, then there is a $f\in C_b(X)$ such that (i)
$0\le f(x)\le 1$ for all $x$, (ii) $f(x)=1$ if and only if
$x\in A$, and (iii) $f(x)=0$ if and only if $x\in
B$. (Hint: try $\rho(x,B)/(\rho(x,A)+\rho(x,B))$.)
- Show that a Cauchy sequence in a metric space
with a convergent subsequence
is necessarily convergent.
|
Friday:
- Work:
- Let $\rho$ and $\sigma$ be metrics on $X$. Show that
$\rho$ and $\sigma$ are equivalent if and only if they have the
same convergent sequences. That is, show that $x_n\to x$ in
$(X,\rho)$ if and only if $x_n\to x$ in $(X,\sigma)$.
- Let $X$ be a metric space. Prove that the uniform limit
of continuous functions $f_n:X\to \mathbf C$ is
continuous.
- Let $X$ be a metric space. Recall that we say $f:X\to
\mathbf C$ is bounded if $\|f\|_\infty<\infty$. A
sequence $(f_n)$ of functions $f_n:X\to\mathbf D$
is uniformly bounded if there is a $M$ such that
$\|f_n\|_\infty\le M$ for all $n$. Also, $(f_n)$ is
called uniformly Cauchy if for all $\epsilon>$ there is
$N$ such that $n,m\ge N$ implies $|f_n(x)-f_m(x)|<\epsilon$
for all $x\in X$. Show that a uniformly Cauchy sequence
$(f_n)$ of bounded functions is uniformly bounded. In
particular, $(f_n)$ converges to a bounded function.
|
