# Is a kernel space necessarily high dimensional?

It's very possible that I'm misunderstanding one or more terms here, so I will just try to explain what I understand (and why it doesn't make sense) :)

Say I have an $N$x$D$ data matrix, i.e. $N$ samples with $D$ features, and I compute an $N$x$N$ kernel out of this data. Now, according to the first answer here, I have computed the similarities in a high dimensional space without explicitly computing where the points lie in this space.

Now, is this necessarily true? What if $D$ > $N$ ? Then I actually would have computed the similarities in a lower dimensional space, right? If not, I would appreciate an explanation why :)

Thanks for any help,