Is it normal to want to be friends after someone has rejected you?

I was recently rejected by someone. They didn't say it outright, but it is obvious they're not interested.

I'm not sure if I should try to be friends with them. I've texted them a couple times and when we talk about things that we're both interested in they tend to reply fairly quickly (Usually within minutes, sometimes within seconds). They're typically not the one to initiate the conversation in texting or person.

I'm not sure if this is because they might feel there's awkward tension between us because they still think I want to pursue them romantically or if they don't want anything to do with me.