A point $x$ of the unit sphere $S(X)$ of the Banach space $X$ is called a multi-smooth point of order $n$ if there exist exactly $n$-independent continuous linear functionals $g_{1}, \ldots , g_{n}$, in $S($ $X^*)$, the unit sphere of the dual of $X$, such that $g_{i}(x)=1$, for $1\leq i\leq n$. The object of this paper is to characterize multi-smooth points of some function and operator spaces.

Simpson's Rule is an accurate numerically stable method of approximating a definite integral using a quadrature with three points, obtained by integrating the unique quadratic that passes through these points. The error term in the method is a function of the fourth derivative of the integrand. Therefore, it is easy to see that the method is exact for cubics, since the fourth derivative of a cubic is zero, and there is no error. The error analysis uses Taylor series. In our simple proof, we will use ordinary integration techniques.

This paper provides an expository survey of the characterizations and properties of four ambient spaces for cones generated by matricial inner products. The partition induced on the vector space of linear operators on complex square matrices is also displayed.