Phy622 Mathematical Methods 3

01-12

Dual Space

Set of all linear homogenous maps from $V \rightarrow F$ is called its dual, or $\tilde V$ with $\langle \alpha a+\beta b;\gamma c + \delta d\rangle = \alpha\gamma\langle a;c\rangle + \beta\gamma\langle b;c\rangle + \alpha\delta\langle a;d\rangle + \beta\delta\langle b;d\rangle$

Hence the dual is also a [01-07#Vector Spaces|Vector Space].

Subspaces

A subset $V’$ of $V$ which is also a Vector space over $F$

Trivially,

• Null Space - $V’ = {|0\rangle}$
• $V$ itself

Other subspaces are called proper subspaces.

• $\mathbb R^2 \subset \mathbb R^3$
• $\mathbb{M}$
  • $\text{Symmetric} \subset \mathbb{M}$
  • $\text{Anti-Symmetric} \subset \mathbb{M}$
  • Hence $\mathbb{M} = S \oplus A$
• Sq Int functions ($L^2$) $\subset$ Functions on $\mathbb R \to \mathbb{R}$

Norm

Map $||w||: V \to \mathbb{R}$ such that:

1. $\forall v \in V, ||v||\ge 0$
2. $||v|| = 0 \iff |v\rangle= |0\rangle$
3. $||\alpha v|| = |\alpha| ||v||$
4. Trinagle Inequality: $||u+v|| \le ||u||+||v||$

A space with a norm becomes a [[01-07#Metric space|Metric Space]] with $d(u, v) = ||u-v||$.

Example

• Euclidean norm
• Taxi Cab Norm on $\mathbb R^n = \sum_i^n ||x_i||$
• pNorm = $(\sum_i^n x_i^p)^\frac{1}{p}$
• max Norm = $\max_i(x_i)$ = $\infty$-norm

Exercises

1. Prove $||u-v|| \ge ||u||-||v||$
2. Prove space with norm is a metric space
3. Prove max norm is $\infty$ Norm