Originally posted by Sandy Hancock Absolutely correct on all three points. The "distortion" you are seeing is due to perspective.
The second image does indeed show extreme barrel distortion, but that is the point of fisheye lenses.
Amen to these very good comments and examples to differentiate between linear and perspective distortion. People often do get the two confused or think they are the same. Getting more into the frame via wide angle lenses does come with a price, and we normally just accept this fact.
A fisheye is a funhouse fun lens having both types of distortion to a very high degree but it also achieves a super-wide angle view. The barrel distortion can be corrected pretty easily post process with the right software, but perspective distortion being a built-in phenomenon for a wide angle view is harder to compensate for. Even if we could position the camera at the right hight and leveling or correct the leaning of vertical subject matter, there remains the front-to-back aspect of perspective distortion. Depth is exaggerated so background objects appear smaller and farther away compared to foreground subjects. The depth of objects like buildings, busses etc. appear elongated.
The only way I've seen to get all of it accurate for a wide angle view is like BigMack suggested- use a "normal" perspective lens, the 43mm Limited on FF or a 28mm lens on APS-C, on a tripod, carefully pan 2 or more shots, and stitch them into a pano.
The DA 15mm and DA 21mm Limited are both excellent and useful lenses, and both remarkably compact and well-made. But they invite a different set of uses. For all-around, the 21mm is far more versatile, with its perspective situation not being so extreme. The 15mm is an amazingly compact and effective ultra-wide, and a wonderful accessory combination lens for other fine lenses, including the 21mm, the DA 20-40mm Limited, or any fine wide-to-tele zoom lens.
The excellent DA 12-24mm f/4 can produce images with even less linear distortion, and better edge performance than the DA 15mm Limited, and when those factors are important, it is a better one to take along. But in terms of a fine accessory wide angle carry-along to expand the wide angle capability of your kit, I pack my amazingly small, beautiful 15mm LTD into the front accessory pouch of my camera belt holster case, with the main lens on-camera in the case! Can't do that with any other ultra wide angle lens!