In the song "Jorden är ett litet rum," Eva Dahlgren sings "kvinnan som lever sitt liv i Stockholm..." Would it be correct to say "kvinnan vem lever sitt liv i Stockholm"? I ask in part because the equivalent in English -- people using that in instances when they could use who -- drives me nuts. For example, one hears people in the U.S. say things like "man that just walked down the street," instead of "man who just walked down the street." Would vem have been a more accurate, or altogether incorrect, choice?
Yes, it would be utterly incorrect to use vem in this case. The default pronoun to use as subject or object in a relative clause in Swedish is som.1
In all the Nordic languages, vem ‘who(m)’ is used in three ways:
as an interrogative pronoun ‘who(m)?’
By far the most common usage; one might say almost the only usage in practical terms.
as the object of a preposition
Only when immediately following the preposition (‘with whom’, ‘to whom’). Unlike in English, this construction is highly formal and stilted; all the Nordic languages vastly prefer that… with (som… med) to with whom (med vem),2 and this usage must be categorised as reasonably rare, especially in the spoken language.
as a pseudo-relative pronoun that includes its own antecedent (archaic)
That is, meaning ‘(s)he who/they who’ or ‘whoever’. In this use, the old, otherwise entirely defunct nominative eho (Danish and Norwegian hvo) is sometimes used, but the whole construction is exceedingly rare, archaic, and stilted. The only example I can think of that has any currency at all is the Danish (and Norwegian, I think?) fixed phrase hvo intet vover, intet vinder ‘he who dares nothing, wins nothing’ or more idiomatically, ‘nothing ventured, nothing gained’. But this expression doesn’t exist in Swedish at all, and I very much doubt anyone would use vem or eho in this way in modern Swedish in any context at all.
Note that there is no use at all of vem as a simple relative pronoun ‘that/who(m)’ in that list. There are some interrogative pronouns, like var ‘where’ and vilket ‘which’, that do also function as relative pronouns, but vem is not one of them.
It is worth noting that vilket, which does function as a simple relative pronoun, is not used like ‘which’ in English: its antecedent is invariably a clause or some type of more or less clearly defined notion, never a simple noun phrase. This is somewhat similar to (but still distinct from) the difference in English between restrictive and non-restrictive usages of that and wh- words (see StoneyB’s comment above).
This is also why only vilket (neuter) is used relatively: the commune vilken would not make sense with reference to an abstract notion. In cases where vilket is used, the simpler relative pronouns may not be used, just like the corresponding cases in English can only take which, never that. Or to put it in different terms: cases where English can use either ‘that’ or ‘which’3 can only take som in Swedish, while cases where English ‘that’ would be ungrammatical must take vilket in Swedish:
The necklace that/which I bought for her birthday turned out to be a fake, which (*that) is a shame.
Halsbandet som (*vilket) jag köpte till hennes födelsedag, visade sig vara falskt, vilket (*som) var synd.
1 It is perhaps worth noting as an aside that where Swedish and Norwegian have only one relative pronoun, som, Danish has two, distinguishing between der (lit. ‘there’) used only as a subject, and som used as either subject or any other constituent.
2 As in English, the default relative pronoun som ‘that’ cannot be used directly following a preposition: *med som ‘*with that’ is as ungrammatical and nonsensical in Swedish as it is in English.
3 I am counting the difference between restrictive and non-restrictive as belonging to the same case type here; so even if one may sometimes be preferable to the other because of the level of restriction inherent in the relative clause, they are still nominally interchangeable.