I think infatuation and lust at first sight are more common than love at first sight, but I definitely believe love at first sight, or at least very early on, can be a thing that happens. All I have to do to reinforce that opinion for myself is look at the picture of Hubby and me gazing lovingly into each other's eyes, as though each of us was the most precious, amazing thing the other had ever seen. A picture which was taken half an hour after we introduced ourselves to each other for the first time. We knew before the first month of our relationship was out that we would eventually combine households and continue moving up the "relationship escalator," and everyone who was at the party where we met has told me they knew that night that he and I were in it for the long haul.
With Woody, it wasn't quite that fast, but we transitioned from friendship to relationship after only about three weeks, and it was only two or three weeks after that that he said he loved me. Took me a bit longer to allow myself to love him, because I was still hurting from the end of my previous relationship, but I said "I love you" to him about a week or so after the first time he said it to me, and acknowledged to both of us that I'd fallen in love with him (for me, "love" and "in love" aren't quite the same thing) about three weeks after that.
With Woody, it wasn't quite that fast, but we transitioned from friendship to relationship after only about three weeks, and it was only two or three weeks after that that he said he loved me. Took me a bit longer to allow myself to love him, because I was still hurting from the end of my previous relationship, but I said "I love you" to him about a week or so after the first time he said it to me, and acknowledged to both of us that I'd fallen in love with him (for me, "love" and "in love" aren't quite the same thing) about three weeks after that.