England international Lallana arrives after a six-year stint at Liverpool, signing a three-year deal with the Seagulls.
Veltman was a part of the hugely successful Ajax side that reached the Champions League semi-finals in 2018 and is a regular in the Netherlands squad. He also signed a three-year contract.
Potter told the club's website, “Adam brings fantastic experience, personality and quality. He's looking forward to starting again with a different project.
“He was at Liverpool a while and had a lot of success, but he's ready to start a new chapter and we're delighted to have him because he will add a lot to us off and on the pitch. It's a really good move for the club and him.
“When you talk about personality, it's deeper than what you might think. It's about understanding how to get points to get up the table and how you should approach the game, believing that you can win.
“At the same time you have to believe that if you don't play well, if you don't work hard and score goals then you will lose. So it's quite a broad thing. Adam has got a good understanding of the league and knows what it's like being a really important member of a successful club."
Veltman also adds a huge amount of know-how, having spent nearly 20 years at Ajax.
“He's an experienced right-back," Potter added.
“He's played for the Netherlands, he's played for Ajax in the Champions League and was part of a really impressive team that they built. He's comfortable on the ball, he's looking for a fresh challenge and we're delighted to have him."