The gym is the perfect environment for sexual equalization from a physical standpoint. In a club or at a party or any other social gathering our masks are on, we can hide deficits more easily with clothing, our inhibitions may be altered due to alcohol, etc. But in the gym all of that is out the window. Oh women attempt to hide their fat butts with sweat shirts tied around their waists, but everyone is sober - in fact we're in a better than normal state of awareness from the exercise - both sexes physical deficits and assets are out in the open. It's social Darwinism at its finest and both sexes strive for attention in some form or another. Guys can only rely on their physical prowess to impress since there's generally no way to judge a man's socio-economic status in the gym. Women are stripped to a primal physical competition where they're judged on their physical form which is men's primary criteria for mate selection. This is why you commonly hear women complain about 'hating' going to the gym; it has less to do with the actual exercise and more to do with an inability to cope with the intense competition present on a level that most are unacustomed to in other social environments. This is why there is such a market for 'women' only health clubs (such as Curves), in fact I've yet to encounter a men's only gym and I'd speculate that this is due to men experiencing far less intimidation in a gym setting.
Sorry, not your question I know, but I get this question a lot from guys, "should I approach in the gym?" and I feel it always needs some prefacing to really understand the incredible advantage, most guys have in the gym in this regard.
So, here's what you do on your next work out: Play with her a little bit, if she's seeking attention, give her just a taste here and there. Don't make the mistake of approaching if she's not giving you IOIs or AIs, women actually do go to the gym in order to get themselves in shape and may or may not want attention while their masks are off. It's the same principle as giving a girl compliments, less is more, but still do it as a treat. Make eye contact and hold it for just a pause longer than you normally would when the opportunity presents itself, but no more. Wait a bit longer, move on to a different machine and see if she pursues. If she does, ignore her. And I mean it, ignore her as if she weren't even there, until you're completely finished with your last set. Then, when the opportunity permits, make eye contact again, but this time smile. DO NOT go out of your way to "conveniently" use the machine next to her, she'll interpret that as an obvious approach and it's game off. Becareful here because this is the test, you have to do a little reading of her, but remember you have the advantge in the gym. Don't smile like Hannibal Lector, smile as if your Mom or sister had just said something funny to you. If all she wanted was your attention, she'll break off and move on to the next exercise. If she's genuinely interested she'll reciprocate with a smile too or some other sign that she's made a connection with you. Now, again, ignore her totally and go back to your workout regimen and put yourself into it. If you've made a connection she'll pursue and you're in, it's all your's at this point. To segue into an introduction, wait until she's on a machine or using some piece of equipment and feign a need to use it too. Suggest that you work in with her on a machine (important - obviously only do this on the only machine or equipment available at that time), then you're free to start the conversation from there. I'd even go so far as to try a 'neg hit' with her too. For instance try saying something like "If you're not done fooling around with that machine, can I work in with you?" Remember a neg hit isn't an insult, it's a challenge to show you're ****y & funny and confident. Keep in mind the whole time, you are at the advantage in the gym.