I'm wondering if it is possible to define a vector space $V$ of all valid probability density functions over $R^n$ (integrable functions in $R^n$ with area equal to 1).
Where the addition of two elements in this vector space, $f + g$, is defined to be their convolution $f*g$.
And multiplication by scalar $af$ is defined as $(af)(x)=\frac{f(x/a)}{a}$.
Does this form a vector space?