Let’s keep this brief. No. Never put a double space after a period. Well, almost never.
Let’s back up. If you were born before the Reagan administration, you may have learned typing on a typewriter (i.e., not a computer). You were taught to put two spaces after a period. And that was correct. A typewriter has only one font, and that font is a monospaced font. That means each character takes up the same horizontal width. So if you typed a series of lines, the characters would line up both horizontally and vertically. This is great if you’re creating a word search.
In order to make the end of a sentence be visually obvious, two spaces were placed after the period. In a monospaced font, a period takes up the same amount of space as a W and is placed within the middle of that space. So at the end of a sentence, that period already has a large amount of white space to its left. And when the period appears after a thin character, such as a lower-case L, the space between the two becomes quite large. And when only one space is placed after a period, the end of sentence becomes less noticeable.
Therefore, many of us were taught to place a second space after a period. Some say even in a monospaced font this is unnecessary. I think those people have a point in that the second space creates an awkward white space that is overkill. But in the balance between readability and design, I’d err on the side of readability in this case.
Enter modern-day computers and typography. Enter kerning. What is kerning? Kerning refers to the spacing between characters, and any half-decent proportional (i.e., not monospaced) font has custom kerning designed into it. Most word processing and design software will allow you to tighten or loosen the kerning. Sometimes you want to do this for design purposes, sometimes you may do this because you need to squeeze your text into a small area, and some fonts just look better with a tiny kerning adjustment (this is often true at larger sizes). But the point is that today’s fonts include characters designed to take up the amount of horizontal space that is the most visually appealing. This includes the period. It is nestled nicely beside the charcter it follows. No need for an additional space afterwards—one is enough.
Looking into the world of website development, we see that HTML paragraph tags don’t even allow for double spaces. If you enter two (or twenty) spaces side by side within a basic paragraph tag, they will be rendered as one space. Sure, you can simulate a double space by adding a non-breaking space character beside a normal space. And many people (and software programs) do just that. But those people are going out of their way to create bad design.
And that’s what it comes down to—design and readability. I worked for a publishing company as a typesetter in the late nineties. There I worked on technical publications that were gone over with a fine-tooth comb—multiple times. Kerning was evaluated on a character-by-character basis. If it was discovered that a font we used consistently placed too much space between the letter O and C, the kerning within the font was updated (using Fontographer, a once-great software package). It’s not so much a matter of whether you’re using one or two spaces, but making sure that the amount of space after a period is visually appealing and conducive to readability.
The fact is, while monospace fonts have their place, they are normally less readable than proportional fonts. Monospace fonts can still be found on your computer today. The most famous is Courier. Should you include two spaces after periods when using Courier? It’s left up to your preference (or your professor’s).