# What are these real-valued functions called?

Let $V\;$ be some normed $\mathbf{R}$-vector space, and $n > 0$ an integer.

First,

What is the name of the the class of mappings $\prod_i f_i(x_i):V\;^n\to\mathbf{R}$ for some $f_1,...,f_n:V\to\mathbf{R}$ ?

(The mappings in this class are ones that can be "factored" into real-valued functions of the individual coordinates.)

Second, setting $f_1 = ... = f_n = f$, define $f\;^{(n)}:V\;^n\to\mathbf{R}$ as

$$f\;^{(n)}(x_1,...,x_n) \equiv \prod_i f(x_i)$$

What is the name of the class of mappings $f:V\to\mathbf{R}$ for which $f\;^{(n)}:V\;^n\to\mathbf{R}$ is spherically symmetric?

(I'm calling a mapping $g:V\;^n\to\mathbf{R}$ spherically symmetric iff $||\mathbf{x}||=||\mathbf{y}||$ implies that $g(\mathbf{x}) = g(\mathbf{y})$.)

Thanks!

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I’d simply call the first batch product mappings. – Brian M. Scott Nov 19 '11 at 19:28
I would call them tensor products. If $V=\mathbb{R}$ then there not too many spherically symmetric tensor products (if I remember correctly there are only constants and the Gaussian). – Dirk Nov 19 '11 at 19:35