In case you haven’t heard yet, Apple has staked a strong claim in the smartphone market.

And it’s a good space to latch onto, as the worldwide smartphone market grew more than 29 percent and the North American smartphone market doubled in the first quarter of 2008 compared to a year ago, according to new research from Gartner.

Apple, Cupertino, Calif., is the No. 3 vendor of smartphones, selling 1.7 million units worldwide to grab a 5.3 percent share of the market, Gartner said. Apple has built that share up from nothing, offering no smartphone in its portfolio until the debut of its iPhone in June, 2007. In the U.S., Apple is the No. 2 vendor, grabbing 20 percent of the market.