The asserted gains in territory are hard to quantify, or verify. In at least one crucial battle in a suburb of Kyiv, where Russian troops had made their closest approach to the capital, brutal street fighting still raged on Thursday and it was not clear that Ukraine had regained any ground.

But even this muddied picture of Ukrainian progress is helpful for the country’s messaging to its citizens, and to the world — that it is taking the fight to a foe with superior numbers and weaponry, and not just hunkering down to play defense. And it underscores the flawed planning and execution that has bedeviled Russian forces from the start, including supply shortages and demoralizing conditions for its soldiers. Those missteps have enabled Ukraine to unexpectedly go on the offensive.

In particular, by preventing Russian troops from capturing Irpin, a suburban town about 12 miles from the center of Kyiv, Ukraine showed that its strategy of sending small units out from the capital to engage the Russians, often in ambushes, has had success, at least for now.