Phy622 Mathematical Methods 3

01-12

Dual Space

Set of all linear homogenous maps from $V\to F$ is called its dual, or $\tilde{V}$ with $⟨\alpha a+\beta b;\gamma c+\delta d⟩=\alpha \gamma ⟨a;c⟩+\beta \gamma ⟨b;c⟩+\alpha \delta ⟨a;d⟩+\beta \delta ⟨b;d⟩$

Hence the dual is also a [Vector Space].

Subspaces

A subset $V^{\prime }$ of $V$ which is also a Vector space over $F$

Trivially,

• Null Space - $V^{\prime }=|0⟩$
• $V$ 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. $\mathrm{\forall }v\in V,||v||\ge 0$
2. $||v||=0⟺|v⟩=|0⟩$
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 = $\underset{i}{max}(x_i)$ = $\mathrm{\infty }$-norm

Exercises

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