Note that “note that” is simply redundant. Note that omitting it entirely does not change the meaning of the sentence. Note that it’s just a bad writing habit. Note that you don’t need it.

@JeffE, But this is not a mathematical context... What is an alternative to this you know? "Note that" has an additional advantage for handwriting; it can be abbreviated as "N.T.".

@GaryMoore N.B. for nota bene would be a more common abbreviation in this context.

"Note that" is a thousand times better than the opposite "It is trivial that".

"Simply redundant" is not necessarily bad. Starting a sentence with a symbol is probably worse. So do this: "Note that X is a solution of the equation." And not this: "X is a solution of the equation."

@GEdgar That's where I use it, too. I'd still prefer to have a short reason instead ("As verified by direct computation, X is a solution" or "By construction, X is a solution"). Note that this mere "filler" role of note that is far from its literal meaning that I'd prefer to express in form of a "Remark: X is a solution" (which again one would rather use if the content of the remark is not really relevant to the ongoing proof)

@DavidRicherby Other ways exist — I prefer "crucially" or such for emphasis, which avoids putting main info in subordinate sentence.

@GEdgar Knuth's I'd use X's name, as I've often heard recommended: "The integer x is a solution of the equation."

@Blaisorblade Sure, but it might be noteworthy without being crucial.

@GEdgar Meh. There's nothing wrong with starting a sentence with a symbol. "The dog is happy" is fine, so what's wrong with "X is a solution"?

@DavidRicherby, I would say it is for readability. There is not only one mathematician suggesting prevention. Paul Halmos, Terry Tao, ...?

@DavidRicherby I think the point is that when scanning a text visually, our brain can get momentarily confused by a mathematical symbol appearing at the beginning of a sentence, especially when the previous sentence ends with a mathematical symbol. "I tossed the frisbee. The dog ran after it" doesn't usually pose much a problem, but "The vector space V has a basis made up of the elements w, x, y. z is an element of V." can be quite confusing and require reading multiple times before you are sure what is being asserted.

@niloderoock note that obviously, it's trivial to simply leave the addition of “note that” as an exercise to the reader.

@DanRomik I agree that's a reason not to start a sentence with a symbol if the previous sentence ended with one. But "I tossed the frizbee. z is an element of V" seems fine to me, albeit something of a non-sequitur. (Your specific example would also be less confusing if you'd written "w, x and y".)

@DavidRicherby agreed. In my view the whole thing about not starting a sentence with a math symbol is a heuristic rather than an absolute rule. There are no absolute rules to good writing, despite people wishing they can algorithmize the process. But for what it's worth, I think it's a good heuristic and applies most of the time.

What is an alternative to this you know? — I use the empty string! There is no reason to emphasize that you are about to say something; just say it. (I avoid "Remark:"for the same reason.) On the very rare occasions that I want to emphasize something counterintuitive, I write "We emphasize that...". Like @Blaisorblade suggests, I name objects (or rewrite) to avoid starting sentences with symbols.

@JeffE since you are taking here a slightly extreme position, can you honestly claim to never use "note that" in your own writing? Or are you willing to acknowledge that there may be times when it's appropriate to use it after all?

@niloderoock: Note that it is trivial to combine the two, so they are not opposites.

@DanRomik The zeroth rule of writing is that any other rule can be broken if it makes the intended meaning clearer. So of course there are times when "note that" is appropriate, just as there are times when it's appropriate to use sentence fragments, symbols at the start of a sentence, "this" as a pronoun, and goto statements in code. But those times should be rare and embarrassing. (In practice, they're less rare and more embarrassing in my own writing than I'd like.)