Lewis Carroll once wrote a story, "Alice's Adventures Underground", which was later enlarged to "Alice in Wonderland". He there, as you surely know, describes how Alice fell in a rabbit hole and had some adventures underground. This adventures, we are told, are nothing real, just a dream, nothing to be taken serious. But this cannot be true: in chapter one of "Alice in Wonderland", there is a first hint:

"She felt that she was dozing off, and had just begun to dream"

This is, of course, not a proper proof that she was not dreaming before dozing off, since one can have a dream within a dream. You certainly remember Tshuang-Tse, the famous follower of Lao-Tse, saying:

"I, Tshung-Tse, once dreamed to be a butterfly, fluttering between here and there, in all its aims a butterfly. I just knew that I followed my moods like a butterfly, and was unconscious about my human nature. Suddenly I awoke; and there I laid: again "me myself". Now I don't know: was I then a man dreaming to be a butterfly, or am I now a butterfly, dreaming to be a man? Between man and butterfly, there is a barrier. Crossing it is called change."

I myself once dreamed to dream and dreamed to wake up. I never dreamed to fall asleep, but how could I conclude that this is impossible? So the quote doesn't proof that Alice wasn't dreaming. But there is another quote in "Through the Looking-Glass", the seventh chapter:

"All this [the speech of the King] was lost on Alice, who was still looking intently along the road, shading her eyes with one hand."

The White King made a speech, and we are told what he was saying ("I only wish I had such eyes, to be able to see Nobody! And at that distance too! Why, it's as much as I can do to see real people, by this light!"), but Alice didn't hear it. So it existed, but not in the perception of Alice. But it is significant for the beings of a dream to exist only in one's perception, and since the White King has an independent existence, he can't be just a being of here dream.
We even know that it is the other way round: not Alice is dreaming about all the people she meets, but she is just a dream of the Red King, as Twedledee in the fourth chapter reveals. So we can answer the question of the last sentence of the last chapter of "Through the Looking Glass".