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}.


\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.


4) Additive inverse.


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


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.


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



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