Ukrainian, Swedish, Tatar and Turkish invaders
Sorry, that's quite insulting. Ukrainian slavs have populated this area for centuries, just think for example about the Kingdom of Halych-Volyn and most of the important cities were founded by Ukrainians. I do not deny that you left your traces there, but still it's our land and we will always be the rightful owners. Lviv is as Polish as Gdánsk is German or Warszawa is Russian – not a single bit.
Excuse my harsh words, but it's a sensitive subject to me.
The history of this region has been plagued by nationalist sentiment for a long time, you are right. Many sources still carry these old traces that demand objective revisions for contemporary times.
That said, these places do have shared history. I think it is ridiculous to say that Lviv's history is solely Ukrainian. It is under Ukrainian administration and there is no reason to change that, but its history is rich and varied. Same thing with Gdańsk, it has much German history and no one should deny that. It is the aggressive nationalist claims of "reclaiming lost land" are pretty much bullshit. That has always led to war and suffering.
This is a Ukrainian source on the subject of Lviv: http://www.wumag.kiev.ua/index2.php?param=pgs20052/58
It, too, points to shared history. That may sound like politically correct crap, and such approaches are often used by liberal sources, however, drawing clear, solid and dividing lines between groups of people who have lived next to each other for a millennium is just as insane. It's like siblings arguing over the line down the center of their bedroom.
About Warsaw being "Russian:" Near where my great-grandmother used to live, there stands an old building that is now a municipal warehouse, but it used to be the Czarist military's supply depot. Whether Polish nationalists like it or not, Warsaw has just a tad of Old Imperial Russia in her.