Barack Obama’s inaugural speech answered a nagging question that much of the world has been debating: Is America too washed up to lead?

After years of Bush unilateralism, and with the U.S. economy in deep trouble, many claim the decline in American influence is unstoppable.

President Obama begs to differ. Yet the vision of U.S. leadership he laid out differs starkly from the style and substance of George W. Bush.

The new president grasps that there is still a hunger abroad for American leadership. That’s evident from a 17-nation poll for the BBC World Service released last week, which found 67 percent expressing hope that Obama’s election would lead to improved relations with the rest of the world.

But the world wants U.S. leadership that differs from that of the Bush years. Thus the key tenet of Obama’s new foreign-policy doctrine is this: We must lead not just with military might, but also with diplomatic alliances and by living up to our democratic values.

“Recall that earlier generations faced down fascism and communism not just with missiles and tanks, but with sturdy alliances and enduring convictions,” Obama said in his address. “They understood that our power alone cannot protect us, nor does it entitle us to do as we please.

“Instead, they knew that our power grows through its prudent use; our security emanates from the justness of our cause, the force of our example, the tempering qualities of humility and restraint.”

One of Obama’s first acts was to announce the planned closure of the Guantanamo military prison. “We reject as false the choice between our safety and our ideals,” he said, “… and we will not give them up for expedience’s sake.”

What was striking in his address was the absence of any grandiose promises, such as those Bush made in his second inaugural to expand freedom and “end tyranny in our world.”

Obama’s approach seems more Kennedy-esque — a promise to befriend any nation seeking “a future of peace and dignity.” He talked of aid to the poor. He promised to work “with old friends and former foes” to lessen the nuclear threat and combat global warming. His pledge to “responsibly leave Iraq” and forge peace in Afghanistan took up only one line.

Obama also refused to view every foreign problem through the prism of a “war on terrorism.” He made a hard-edged pledge to fight those “who seek to advance their aims by inducing terror and slaughtering innocents,” and promised, “You cannot outlast us, and we will defeat you.”

But he specified that the “war” we are fighting is against “a network of violence and hatred.” In other words, this struggle is against particular groups, not Islamic civilization.

Indeed, he added: “To the Muslim world, we seek a new way forward, based on mutual interest and mutual respect.” However, he warned that those leaders who “blame their societies’ ills on the West” are “on the wrong side of history.” Even after this warning, however, he added “we will extend a hand if you are willing to unclench your fist.”

Many say Obama can’t possibly live up to the world’s high expectations. Skeptics predict his hopes for smart diplomacy will run into the recalcitrance of Iran.

But his address showed a man in tune with the times. We live today not in the unipolar world of the 1990s, but in a non-polar world. The United States has been bogged down in Iraq and Afghanistan, and unwilling to lead on key issues such as global warming. Washington has seemed incapable of leading the way out of the global recession as its own financial crisis worsens and infects others.

Yet rising powers such as India and China aren’t ready to lead on international problems. Europe, united in its economy, is too divided politically to take up that role.

The world seems adrift. In such a power vacuum, extremist groups flourish. People everywhere, including here, feel they have lost control of their future.

Obama recognizes that the world wants him to lead. He knows that his personal story — and what it says about America — give him and his country a new image. He grasps that publics abroad are hoping, even as they doubt, that he can fashion some rescue for the global economy.

So when Obama said “we are ready to lead once more,” he was emphasizing leadership with a difference. His global honeymoon may last longer than many expect.