Relation

Subset of product set $R \subset A \times B$

Function

Relation on $A \times B$, such that A related to B. It can be onto / one-to-one / bijection.

Convergence

We know the cauchy sequence, 1.2, 1.22, 1.222, ... . But Does this converge? To generalize this concept, we have to introduct Metric space.

Metric Space

It has the distance function. with triangle inequality.

It has the concept of Convergence which means $\forall \epsilon>0, ~~ \exists N: n \geq N \implies d(x_n, x) < \epsilon$

Similarly, cauchy sequence means $\forall \epsilon > 0, ~~ M \geq N, \implies d(x_n, x) < \epsilon$. This is weaker concept than convergence and it has no reference point like convergence.

In complete metric spaces, cauchy sequence converges as well.

For example, $1, 1.41, 1.414, ... \rightarrow \sqrt{2}$ is a cachy sequency in $\mathbb{Q}$ but converges in $\mathbb{R}$

Vector Space

Metric space gives abstract space for distance. Similarly, this is abstract space for vector operation.

It has $\times : F \times V \rightarrow V$ (F is called field here) It also has $+ : V + V \rightarrow V$

Normed Space Vector + Metric Space (with "norm" distance function)

norm funcion $\parallel \cdot \parallel$: $V \rightarrow \mathbb{R}_+$

Norm makes V into special metric space since $d(u, v) = \parallel u-v \parallel$ satisfies axioms of metric space.

$\boxed{\text{Complete norm space is called Banach Space.}}$

Inner Product Space Vector Space with inner product

$<\cdot, \cdot>: V \times V \rightarrow \mathbb{R}$

This is a special case of normed space $\implies \parallel u \parallel = \sqrt{<u, v>}$

$\boxed{\text{Complete inner product space is called Hilbert Space.}}$

Inner product space adds the concept of angles.

$\frac{<u, v>}{\parallel u \parallel \parallel v \parallel} = cos \phi$