0
votes
1answer
64 views

Proving the formula for the directional derivatives of the of the sum and dot product of two functions

Define the directional derivative of a function $\textbf{f}$ at $\textbf{c}$ in the direction $\textbf{u}$ by $$\textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) = \lim_{h \rightarrow 0} ...
1
vote
3answers
338 views

set theoretic function, products of sets (product versus Cartesian product)

Regarding the products of functions in axiomatic set theory, two textbooks which I am reading (Halmos; Hrbacek/Jech) have said the following: "There is a natural one-to-one correspondence between ...
3
votes
1answer
231 views

Is $\prod_{\mathbb{R}}\mathbb{R} = \mathbb{R}^\mathbb{R}$?

(If the title is unclear, I'm looking at infinite cartesian product of $\mathbb{R}$ indexed by $\mathbb{R}$.) I thought that I had reasoned this rather well, as follows: $\mathbb{R}^\mathbb{R} = ...