Vector space of functions

As mentioned in the previous article, a vector space $\textit{V}$ is a set of objects that follows certain rules of addition and multiplication. This implies that a set of functions $\textit{V}=\left \{ f(x),g(x),h(x)\cdots \right \}$ that follows the same rules, forms a vector space of functions. The properties of a vector space of functions are:

1) Commutative and associative addition for all functions of the closed set $\textit{V}$ .

$f(x)+g(x)=g(x)+h(x)$

$\left [f(x)+g(x)\right ]+h(x)=f(x)+\left [g(x)+h(x)\right ]$

2) Associativity and distributivity of scalar multiplication for all functions of the closed set.

$\gamma\left [ \delta f(x) \right ]=(\gamma\delta)f(x)$

$\gamma\left [ f(x)+g(x) \right ]=\gamma f(x)+\gamma g(x)$

$(\gamma +\delta)f(x)=\gamma f(x)+\delta f(x)$

where $\gamma$ and $\delta$ are scalars.

3) Scalar multiplication identity.

$1f(x)=f(x)$

4) Additive inverse.

$f(x)+[-f(x)]=0$

5) Existence of null vector $\boldsymbol{\mathit{0}}$ , such that

$\boldsymbol{\mathit{0}}+f(x)=f(x)$

where $\boldsymbol{\mathit{0}}$ in this case is a zero function that returns zero for any inputs.

Similarly, a set of linearly independent functions $y_0(x),y_1(x),\cdots,y_n(x)$ forms a set of basis functions. We have a complete set of basis functions if any well-behaved function in the domain $a\leq x\leq b$ can be written as a linear combination of these basis functions, i.e.

$f(x)=\sum_{n=0}^{\infty}c_ny_n(x)$

In quantum chemistry, physical states of a system are expressed as functions called wavefunctions.

Vector space of functions

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