The set of all [[Square integrable functions]], $f(x)$ or $f(z)$ (i.e. [elementary functions](Analysis%20(index).md#Functions) that may take real or complex arguments) defined over a finite interval, $I$, is referred to as $L^2(I)$ where the label $L^2$ refers to the fact that they are [square integrable](Square%20integrable%20functions.md). This is generalized by [function Spaces.](Function%20spaces.md)
# Properties
These [properties](Vector%20spaces.md#Properties) are rewritten as follows for functions:
1. We can write $a(f+g)(x)=af(x)+ag(x)$ where $a\in\mathbb{R}$
2. $a(bf(x))=b(af(x))$ where $a, b \in\mathbb{R}$
3. There exists $f(x)=0$ in $L^2(I)$.
In addition,
4. $L^2(I)$ is an [[Inner products]] space where $\langle f,g \rangle = \int_I dxf^*(x)g(x)$
and the naming $L^2(I)$ indicates that,
5. $\langle f,f \rangle = \int_I dx|f(x)|^2<\infty$ ([square integrability](Square%20integrable%20functions.md))
# Vector representation
[Square integrable functions](Square%20integrable%20functions.md) are vectors.
The way functions are represented as vectors (as one would need to on a computer), is that they are discretized as $N$ component vectors. Thus a function may be _approximated_ as a vector, $\mathbf{f}=(f_1...f_N)$. Where the interval is also discretized into a large number of small intervals of width $\tau/N$. Each vector component is parameterized by $t$ and located in an interval, $\tau$ where $\tau$ is centered on points $t=t_i$. For a [Continuous function](Continuous%20function.md) $N=\infty$.
(Images adapted from Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press (2019))
# Functional Space
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# Recommended Reading
For an introduction to $L^2(I)$ spaces see:
* [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) pgs. 26-28. Presented here is an introduction to the notion that _functions are also vectors_ as well as the notion of certain commonly encountered functions belonging in $L^2(I).$
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces
#MathematicalFoundations/Analysis/FunctionalAnalysis