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Mathematical Analysis

Sam Macdonald

Section 4.1 Topology and Metric Spaces

Subsection Inner Products and Cauchy-Schwarz

“Keep in mind, measurement is not just in numbers, but stories.”
―Pearl Zhu

Definition 4.1. \(C^n\) and \(R^n\).

Set \(\mathbb{C}^n=\left \{\begin{bmatrix}x_1 \\ x^2 \\ \vdots \\ x_n \end{bmatrix}\,:\, x_i\in\mathbb{C}\right\}\) and \(\mathbb{R}^n=\left\{\begin{bmatrix}x_1\\x^2\\ \vdots\\x_n\end{bmatrix}\,:\,x_i\in\mathbb{R}\right\}\) (\(n\geq 1\))

Convention 4.2.

We'll often write \((x_1,\dots,x_n)\) in place of \(\begin{bmatrix}x_1\\x^2\\ \vdots\\x_n\end{bmatrix}\) and \(x\in\mathbb{C}^n\) in place of \(\vec{x}\in\mathbb{C}^n\)

Definition 4.3. Inner Product, Norm.

The inner product on \(\mathbb{C}^n\) is given by \(\langle{x,y}\rangle=\sum_{i=1}^nx_i\bar{y_i}\,\forall\,x,y\in\mathbb{C}^n\text{.}\) Define the norm on \(\mathbb{C}^n\) by \(\left|{\left|{{x}}\right|}\right|=\langle{x,x}\rangle^{\frac{1}{2}}=(\sum_{i=1}^n\left|{x_i}\right|^2)^{\frac{1}{2}}\text{.}\)

Convention 4.4.

Sometimes people define \(\langle{x,y}\rangle=\sum_{i=1}^n\bar{x_i}y_i\text{.}\) (especially physicists)

Remark 4.6.

Sesquilinearity means “1 and a half linear.” In real case it's bilinear.
\((a-b)^2\geq 0\rightarrow a^2-2ab+b^2\geq 0\rightarrow ab\leq \frac{1}{2}(a^2+b^2)\)
\begin{equation*} \begin{aligned} \left|{\langle{x,y}\rangle}\right| &=\left|{\sum_{i=1}^nx_i\bar{y_i}}\right|\\ &\leq \sum_{i=1}^n\left|{x_i}\right|\cdot{\left|{y_i}\right|}\\ &\leq \sum_{i=1}^n\frac{1}{2}(\left|{x_i}\right|^2+\left|{y_i}\right|^2)\\ &=\frac{1}{2}(\left|{\left|{{x}}\right|}\right|^2+\left|{\left|{{y}}\right|}\right|^2) \end{aligned} \end{equation*}
If \(x,y\neq 0\) then
\begin{equation*} \left|{\langle{\frac{x}{\left|{\left|{{x}}\right|}\right|},\frac{y}{\left|{\left|{{y}}\right|}\right|}}\rangle}\right|\leq \frac{1}{2}(\left|{\left|{{\frac{x}{\left|{\left|{{x}}\right|}\right|}}}\right|}\right|^2+\left|{\left|{{\frac{y}{\left|{\left|{{y}}\right|}\right|}}}\right|}\right|^2=\frac{1}{2}(1^2+1^2)=1. \end{equation*}
Multiply by \(\left|{\left|{{x}}\right|}\right|\left|{\left|{{y}}\right|}\right|\) to get \(\left|{\langle{x,y}\rangle}\right|\leq \left|{\left|{{x}}\right|}\right|\cdot\left|{\left|{{y}}\right|}\right|\text{.}\)
If \(x=0\) or \(y=0\) then \(\left|{\langle{x,y}\rangle}\right|=0=\left|{\left|{{x}}\right|}\right|\cdot\left|{\left|{{y}}\right|}\right|\)
\begin{equation*} \begin{aligned} \left|{\left|{{x+y}}\right|}\right|^2 &=\langle{x+y,x+y}\rangle\\ &=\langle{x,x}\rangle+\langle{x,y}\rangle+\langle{y,x}\rangle+\langle{y,y}\rangle\\ &=\left|{\left|{{x}}\right|}\right|^2+\langle{x,y}\rangle+\overline{\langle{x,y}\rangle}+\left|{\left|{{y}}\right|}\right|^2\\ &=\left|{\left|{{x}}\right|}\right|^2+2\text{Re}(\langle{x,y}\rangle)+\left|{\left|{{y}}\right|}\right|^2 \end{aligned} \end{equation*}

Subsection Metric Spaces

We can use the norm to define a notion of distance: the distance from \(x\) to \(y\) is \(\left|{\left|{{x-y}}\right|}\right|\text{.}\)

Definition 4.12.

Let \(X\) be a set. A metric on \(X\) is a function \(d:X\times X\rightarrow\mathbb{R}_+\) such that \(\forall\,x,y,z\in X\)
  1. Positive Definite.
    \(\displaystyle d(x,y)=0\iff x=y\)
  2. Symmetric.
    \(\displaystyle d(x,y)=d(y,x)\)
  3. Triangle Inequality.
    \(\displaystyle d(x,z)\leq d(x,y)+d(y,z)\)
A metric space is a pair \((X,d)\) where \(X\) is a set and \(d\) is a metric on \(X\text{.}\) We'll typically write \(X\) for a metric space where the metric is understood.

Example 4.13. Euclidean Metric.

The pair (R, d) is a metric space, where d is defined by d(x, y) = x − y. The metric d is called the usual (or absolute value or Euclidean) metric for R. The space R is understood to have the usual metric unless otherwise specified.

Example 4.14. Discrete Metric.

For a set \(X\text{,}\) the discrete metric on \(X\) is \(d{X\times X}\rightarrow\mathbb{R}_+\text{.}\)
\begin{equation*} \begin{aligned} d(x,y)=\begin{cases} 1 &\ \text{ if } x\neq y\\ 0 &\ \text{ if } x=y \end{cases} \end{aligned} \end{equation*}
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