BANACH SPACES

 

Banach spaces[edit]

Main article: Banach space

Banach spaces, introduced by Stefan Banach, are complete normed vector spaces.^[48]

A first example is the vector space ${\ell }^{�}$ consisting of infinite vectors with real entries ${\displaystyle \mathbf{�}=\left({�}_1,{�}_2,\dots ,{�}_{�},\dots \right)}$ whose $�$-norm ${\displaystyle (1\le �\le \mathrm{\infty })}$ given by

$${\displaystyle \Vert \mathbf{�}{\Vert }_{\mathrm{\infty }}:=\underset{�}{sup}|{�}_{�}|\text{ for }�=\mathrm{\infty },\text{ and }}$$

The topologies on the infinite-dimensional space ${\ell }^{�}$ are inequivalent for different $�.$ For example, the sequence of vectors ${\displaystyle {\mathbf{�}}_{�}=\left(2^{-�},2^{-�},\dots ,2^{-�},0,0,\dots \right),}$ in which the first $2^{�}$ components are $2^{-�}$ and the following ones are ${\displaystyle 0,}$ converges to the zero vector for ${\displaystyle �=\mathrm{\infty },}$ but does not for ${\displaystyle �=1:}$

$${\displaystyle \Vert {\mathbf{�}}_{�}{\Vert }_{\mathrm{\infty }}=sup(2^{-�},0)=2^{-�}\to 0,}$$

but

$${\displaystyle \Vert {\mathbf{�}}_{�}{\Vert }_1=\sum _{�=1}^{2^{�}}{2}^{-�}=2^{�}\cdot 2^{-�}=1.}$$

More generally than sequences of real numbers, functions ${\displaystyle �:\mathrm{\Omega }\to \mathbb{�}}$ are endowed with a norm that replaces the above sum by the Lebesgue integral

$${\displaystyle \Vert �{\Vert }_{�}:={\left({\int }_{\mathrm{\Omega }}|�(�){|}^{�}��(�)\right)}^{\frac{1}{�}}.}$$

The space of integrable functions on a given domain $\mathrm{\Omega }$ (for example an interval) satisfying  and equipped with this norm are called Lebesgue spaces, denoted ${\displaystyle {�}^{�}(\mathrm{\Omega }).}$^{[nb 9]}

These spaces are complete.^[49] (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.^{[nb 10]}) Concretely this means that for any sequence of Lebesgue-integrable functions ${\displaystyle {�}_1,{�}_2,\dots ,{�}_{�},\dots }$ with  satisfying the condition

$${\displaystyle \underset{�,�\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{\left|{�}_{�}(�)-{�}_{�}(�)\right|}^{�}��(�)=0}$$

there exists a function $�(�)$ belonging to the vector space ${\displaystyle {�}^{�}(\mathrm{\Omega })}$ such that

$${\displaystyle \underset{�\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }}{\left|�(�)-{�}_{�}(�)\right|}^{�}��(�)=0.}$$

Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.^[50]