Lecture 02 - Banach Spaces

#math

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A hierarchy of mathematical spaces: The inner product induces a norm. The norm induces a metric. The metric induces a topology. We are working from outside in, starting from metric spaces trying to work towards Inner product spaces. We are seeking inner products since they have the ability to determine whether two objects in the space are orthogonal to each other.

Consider the following wave equation problem:
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\begin{aligned} y_{x x} & = \frac{1}{k^{2}} y_{t t} \end{aligned}

With initial conditions:

\begin{aligned} y (0, t) & = y (π, t) = 0 \\ y (x, 0) & = F (x) \\ y_{t} (x, 0) & = 0 \end{aligned}

We know, due to Bernoulli, we can solve this problem using separation of variables:

y (x, t) = f (x) g (t)

Proceeding forward by differentiating we can obtain:

f^{″} g = \frac{1}{k^{2}} f g^{″} ⟺ \frac{f^{″}}{f} = \frac{1}{k^{2}} \frac{g^{″}}{g}

Since the left hand side is a function of $x$ only while the right hand side is a function of $t$ only then we deduce that they must be equal to a constant for all $x$ and $t$ . Thus we obtain the following two ODE problems:

\begin{aligned} f^{″} (x) & = c f (x) f (0) = f (π) = 0 \\ g^{″} (t) & = c k^{2} g (t) g^{'} (0) = 0 g (0) = F (x) / f (x) \end{aligned}

Thus we reduced our PDE problem to a system of ODEs. We obtain the following solutions:

\begin{aligned} If c = - n^{2} ⟶ & f (x) = s i n (n x) \\ g (t) = a_{n} c o s (k n t) + b_{η} s i n (k n t) \end{aligned}

Thus we obtain:

y_{n} (x, t) = s i n (n x) (a_{n} c o s (k n t) + b_{n} s i n (k n t))

and by the super position principle our solution can be written as a series of $y_{n}$ . Thus we obtain our solution to be:

y (x, t) = \sum_{n = 1}^{\infty} s i n (n x) (a_{n} c o s (k n t) + b_{n} s i n (k n t))

Banach Space

Definition

A metric (or a distance function) $d$ on a a set $M$ is a function on $f : M \times M \to R$ that satisfies the following conditions:

Non negativity: $d (x, y) \geq 0$ and $d (x, y) = 0 ⟺ x = y$
Symmetry: $d (x, y) = d (y, x)$
Triangle Inequality: $d (x, z) \leq d (x, y) + d (y, z)$

Exercise

Let $M$ be the set of cities in Indonesia. Determine which of the following are a metric on $M$ .

$d (A, B)$ is the price of 2nd class railway ticket from $A$ to $B$ .
$d (A, B)$ is the offpeak driving time from $A$ to $B$ .

Definition

Let $V$ be a real (or complex) vector space. A norm on $V$ is a real-valued function, $| | X | |$ , such that:

$| | X | | \geq 0$ and $| | X | | = 0 ⟺ X = 0$
$| | λ X | | = | λ | | | X | |$ for $λ \in R$ (or $C$ ) and $v \in V$
$| | X + Y | | \leq | | X | | + | | Y | |$

Definition

We say a sequence ${x_{n}}$ is convergent in a metric space $(M, d)$ if $\forall ϵ > 0, \exists N >> 1$ such that $\forall n \geq N$

d (x_{n}, x) < ϵ

Let ${x_{k}}_{k \geq 1}$ in a metric space $(M, d)$ is a Cauchy sequence if $\forall ϵ > 0 \exists N \in N$ such that if $k, l > n$ then $d (x_{k}, x_{l}) < ϵ$
$(M, d)$ is a complete metric space if every Cauchy sequence is convergent.

Definition

A Banach space is a complete normed space.

Example

$l_{2}^{n}$ on $R^{n}$ or $C^{n}$ is the set of sequences of length n, i.e. $x \in l_{s}^{n}$ then $x = (x_{1}, x_{2}, . . ., x_{n})$ such that

| | x | | = \sum_{i = 1}^{n} {x_{i}}^{2}

$Ξ = C ([a, b])$ $| | f | |_{\infty} = \underset{a \leq x \leq b}{M a x} | f (x) |$