What is Functional Analysis

1. Functional analysis is analysis (mathematical analysis or advanced algebra) on infinite-dimensional spaces.
2. Origin (background): Specific analytical problems in mathematical physics.
3. Research method: Abstract specific analytical problems into problems of analysis on spaces with topological (metric) and/or algebraic structures.
4. Current status: Rich in content, widely applied, an essential tool for mathematicians.

Calculus & Linear Algebra	Finite-dimensional $\mathbb{R}^n (\mathbb{C}^n)$	Matrices and Functions	Eigenvalues	Zeros
Functional Analysis	Infinite-dimensional spaces	$l^p$ , $C[a,b]$	Operators	Spectrum

Incomplete Normed Linear Spaces

1. Polynomials $P[0,1]$, $|P|{P[0,1]}:=\max{t\in[0,1]}{

   P(t)

   }$, its completion is $C[0,1]$.

2. $|f|{C[0,1]}:=\int^1{0}

   f

   \,dx$, its completion is $L^1[0,1]$.

Definition of Operator Norm

\[\|T\|_{\mathcal{L}(\mathcal X,\ \mathcal Y)}=\sup_{x\in\mathcal X\backslash\{0\} }\frac{\|Tx\|}{\|x\|}\]