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43:  Complete Metric Spaces

$(X, d)$ is a metric space.

Defn:  $x_n$ is Cauchy if for all $\epsilon > 0$, there exists an $N$ such that for all $n, m > N$, 
$d(x_n, x_m) < \epsilon$.

Defn:  $(X, d)$ is complete if every Cauchy sequence in $X$ converges in $X$. 

Lemma:  convergent implies Cauchy

Lemma:  $(X, d)$ complete, $A$ closed in $X$ implies $(A, d)$ is complete. 

Lemma:  $(X, d)$ complete if and only if $(X, \overline{d})$ is complete where 
$\overline{d}(x, y) = min\{d(x, y), 1\}$.

Lemma 43.1:  $(X, d)$ is complete if every Cauchy sequence has a convergent subsequence.

Lemma:  A Cauchy sequence is bounded.
\u
Thm 43.2:  $R^k$ is complete in both the euclidean metric $d$ or the square metric $\rho$.

Lemma 43.3:  ${\bf x_n} \rightarrow {\bf x}$ in $\Pi X_\alpha$ if and only if 

\centerline{$\pi_\alpha({\bf x_n})
\rightarrow \pi_\alpha({\bf x})$}

Thm 43.4:  ${\cal R}^\omega$ is complete with respect to 

\centerline{$D({\bf x}, {\bf y}) =
sup\{{\overline{d}(x_i, y_i) \over i}\}$}


Recall $Y^J = 
\{(y_\alpha)_{\alpha \in J}\}= \{f: J \rightarrow Y\}$ 
\vskip 5pt
\centerline{where $f(\alpha) = y_\alpha$}

Thus, 
$\overline{\rho}({\bf x}, {\bf y}) = sup\{\overline{d}(x_\alpha, y_\alpha) \}$ is the same as

\centerline{$\overline{\rho}(f, g) = sup\{\overline{d}(f(\alpha), g(\alpha) \}$}

Thm 43.5:  $(Y, d)$ complete implies $Y^J$ is complete with respect to the uniform metric.

Recall $f$ bounded if $f(X)$ is bounded
\hfil \break
 [there exists $B(x_0, r)$ such that $f(X) \subset B(x_0, r)$].

Defn:  ${\cal B}(X, Y) = \{f: J \rightarrow Y ~|~ f$ bounded $\}$.

If $X$ is a topological space, define 

\centerline{${\cal C}(X, Y) = \{f: J \rightarrow Y ~|~ f$ continuous $\}$.}

Thm 43.6:  ${\cal C}(X, Y)$ and ${\cal B}(X, Y)$ are closed subsets of $Y^X$ under the uniform
topology.  Thus if $Y$ is complete in uniform metric, ${\cal C}(X, Y)$ and ${\cal B}(X, Y)$ are
complete.

Defn:  The sup metric, ${\rho}(f, g) = sup\{{d}(f(\alpha), g(\alpha) \}$, is a metric on ${\cal B}(X, Y)$

Note:  $\overline{\rho}(f, g) = $ min$\{ {\rho}(f, g), 1\}$.

Note:  If $X$ compact, $Y^X = {\cal B}(X, Y)$

Thm 43.7:  There is an isometric imbedding of $(X, d)$ into a complete metric space.



Defn:    If $h: X \rightarrow Y$ is an isometric imbedding of metric space $X$ into complete metric space $Y$, then $\overline{h(X)}$ is a complete metric space called the completion of $X$.

The completion of $X$ is uniquely determined up to isometry.



HW  33: 1; 43: 8, 10

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44:  There exists a continuous surjective function 

\centerline{$f: [0, 1] \rightarrow [0, 1] \times [0, 1]$.}


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The $ij^{th}$ entry of $P^k$ is the probability that you are in the jth state after exactly $k$ steps given that you started in the ith state.

Suppose that $p_i$ is the probability that you start in state $i$.  Let $p = (p_1, ..., p_n)$.  Then $pP^k = (s_1, ..., s_n)$ where $s_j$ is the probability that you are in the jth state after exactly $k$ steps.


If $P = \left(\matrix{I & 0 \cr R & Q}\right)$, then can use $Q^k$ instead of $P^k$.

{\bf If there are transient states:}

If $Q$ represents transient states, then $lim_{n \rightarrow \infty}Q^n} = 0$

If $N = (I - Q)^{-1} = \Sigma_{n=0}^\infty$, then the $ij^{th}$ entry of $N$ is the expected number of times you are in state $j$ given that you started in state $i$.

Hence the expected number of steps before absorption is the sum of the ith row of $(I - Q)^{-1} $ given that you started in state $i$.

If $B = NR = (I - Q)^{-1}R$, then $b_{ij} is the probability that absorbed in state $j$ given that you started in state $i$.

{\bf Ergodic (there are NO transient states):}

{\it If regular (i.e. $P^k$ is a positive matrix for some $k$):}

$P^n$ is a positive matrix for all large $n$ ($n \geq k$).

$lim_{n \rightarrow \infty}P^n} = W = \left(\matrix{{\bf w} \cr . \cr . \cr . \cr {\bf w} }\right)$

$lim_{n \rightarrow \infty}pP^n = w}

$wP = w$

If $E = (I - Z + JZ_{dg})D$, then $e_{ij}$ is the expected number of steps from state $i$ to state $j$ (without going through state $j$ in between, i.e., first time getting to/returning to state $j$).

$ e_{ii} = {1 \over w_i}$

Regular if and only if period = 1.

{\it If not regular}

Period > 1

Fix i:  d = Period = gcd$\{ n ~|~ $ there is a path from $u_i$ to $u_i$ of length $n \}$

States can be partitioned into $d$ periodic classes, $C_0, ..., C_{d-1}$ such that if you start at a vertex in $C_i$, then after $k$ steps, you are in a vertex in $C_{i + k (mod d)}.






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