I realize this is probably very specific to the person writing the textbook, but I was wondering if anyone else out there knows the answer.

I encountered this in one of my textbooks: $\mathbb{R}_{++}$. Now, I know that $\mathbb{R}_+$ is used to denote the set of positive real numbers, but I've never encountered the former before. The book uses both in the same section, so clearly they do not mean the same thing...


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    $\begingroup$ If the book does not define it, and it isn't a typo, I'm not sure what it could mean. Can you tell us what book? $\endgroup$ – Thomas Andrews Apr 10 '13 at 19:24
  • $\begingroup$ Convex Optimization - Stephen Boyd. It is on page 640 (Appendix A) There's a pdf of it available here: stanford.edu/~boyd/cvxbook $\endgroup$ – user71959 Apr 10 '13 at 19:26
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    $\begingroup$ @user71959 On page $711$ of the pdf you can see that $\Bbb R_{++}$ are the positive real numbers while $\Bbb R_+$ are the nonnegative real numbers. $\endgroup$ – Git Gud Apr 10 '13 at 19:29
  • $\begingroup$ @GitGud Actually, I think they are using $\mathbb R_+$ for "non-negative." The text says: $\mathbb R_+,\mathbb R_{++}$ are "Nonnegative, positive real numbers." This coincides with the definition of $S^n_{+}$ as positive semi-definite and $S^n_{++}$ as positive definite. (Really hate that they use $S^n$ for symmetric matrices, but there it is.) $\endgroup$ – Thomas Andrews Apr 10 '13 at 19:33
  • $\begingroup$ @ThomasAndrews I agree, I noticed it and edited soon after posting the comment. If you want to post it an answer with that analogy with the symmetric matrices, I will upvote. Better this doesn't show as unanswered. $\endgroup$ – Git Gud Apr 10 '13 at 19:34

From GitGud's search, it appear that $\mathbb R_+$ is the non-negatives, and $\mathbb R_{++}$ are the positive. That is $\mathbb R_+$ includes zero.

This is analogous to their definition of $\mathbb S^n$ as symmetric matrices, $\mathbb S^n_+$ as the symmetric positive semi-definite, and $\mathbb S^n_{++}$ as the symmetric positive definite matrices.

So $\mathbb R_{+}$ means "positive, sort of." And $\mathbb R_{++}$ means "no, I really mean it."

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  • $\begingroup$ I had already promised I'd upvote, but still: +$1$ for the humor. $\endgroup$ – Git Gud Apr 10 '13 at 19:42
  • $\begingroup$ I was thinking of Orwell's Newspeak: "double-plus good." @GitGud $\endgroup$ – Thomas Andrews Apr 10 '13 at 19:49
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    $\begingroup$ It is real inflation... $\endgroup$ – copper.hat Apr 10 '13 at 20:13

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