The end to Edinburgh's European adventure finishes any competition for honours for another season as they sit second bottom of the RaboDirect PRO12 league.
But Bradley was encouraged by his team's travails in the Heineken Cup and insisted they would look to add silverware to plaudits next year.
"There are areas we have to improve. There is young talent coming through and quality international players with a burning desire to win something," said Bradley.
"Edinburgh Rugby has taken a positive step forward."
Bradley's side competed well and even when they looked out of the game after Pienaar's 76th-minute penalty which opened a ten-point gap, Jim Thompson's try put them within touching distance of the final.
And Bradley admitted he was left with mixed feelings after his side played some attractive rugby but were unable to turn their dominant periods into consistent points.
"We are disappointed and a little bit frustrated," he added.
"We had enough opportunities to build a bigger points total than we did.
"In the second half Ulster shut us down a little bit more. They stifled our ability to play attacking rugby.
"Ulster got the upper hand in the scrum but we had chances to win the game."
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