Hvort sem herbergið er lítið eða stórt, vil ég ekki henda góðum bókum.

It is a fixed Icelandic pattern meaning something like whether ... or ... or regardless of whether ... or ....

So the first clause presents two possibilities:

• hvort sem herbergið er lítið eða stórt

and the main clause stays true in either case.

This is a very common structure, and sem is part of the expression, not a separate word you can just drop.

Because hvort sem introduces a subordinate clause.

In Icelandic, subordinate clauses usually have more regular subject + verb order:

• herbergið er lítið

rather than main-clause-style verb-second inversion.

So this is a good contrast:

• subordinate clause: herbergið er
• main clause after fronting: vil ég

Icelandic usually puts the definite article on the end of the noun.

• herbergi = room
• herbergið = the room

Here herbergið is the subject of the clause, so it is in the nominative singular. Since herbergi is a neuter noun, the definite ending is -ið here.

Because they agree with herbergið, which is:

• neuter
• singular
• nominative

After vera = to be, predicate adjectives agree with the subject:

• herbergið er lítið
• herbergið er stórt

That -t ending is the normal neuter singular form here.

Because these are predicate adjectives, not attributive adjectives.

Compare:

• stóra herbergið = the big room
  
  • adjective comes before the noun
  • this is attributive
  • weak form is used

but:

• herbergið er stórt = the room is big
  
  • adjective comes after er
  • this is predicate use
  • strong form is used

So even though herbergið is definite, the correct predicate forms are lítið and stórt.

This is because Icelandic is a verb-second language in main clauses.

The whole fronted clause:

• Hvort sem herbergið er lítið eða stórt

takes the first position in the sentence. That forces the finite verb of the main clause into second position:

• vil

and the subject comes after it:

• vil ég

So:

• Hvort sem ... , vil ég ...

If there were no fronted clause, you would normally say:

• Ég vil ekki henda góðum bókum.

Because vilja often takes a bare infinitive.

So Icelandic says:

• vil henda = want to throw away

not:

• vil að henda

This is similar to how English modals work with a bare infinitive, although vilja is not identical to English modal verbs in every way.

In a clause with a finite verb plus an infinitive, ekki normally comes before the infinitive.

So in a neutral clause:

• Ég vil ekki henda góðum bókum.

After fronting and verb-second inversion, the finite verb and subject change places:

• vil ég ekki henda góðum bókum

but ekki still comes before henda.

A useful pattern is:

• finite verb + subject + ekki
  • infinitive

in this kind of sentence.

Because henda takes a dative object in this meaning.

So the thing being thrown away is in the dative:

• bókum = dative plural of bók
• góðum agrees with bókum, so it is also dative plural

Compare:

• nominative plural: góðar bækur
• dative plural: góðum bókum

This is one of the most important things to notice in the sentence.

Here it means to throw away, to discard, or to get rid of.

In context, ég vil ekki henda góðum bókum means not wanting to throw good books away.

Learners often notice that Icelandic has several verbs related to throwing. In this sentence, the important point is that henda is the verb chosen, and it governs the dative.

Yes. You could also say:

• Ég vil ekki henda góðum bókum, hvort sem herbergið er lítið eða stórt.

That would still be grammatical.

The version with Hvort sem ... first gives the condition or contrast up front. It sounds a bit more like setting the scene first, then giving the main point.

The grammar changes slightly in the main clause because of word order:

• with fronting: vil ég
• without fronting: ég vil