Har du ikke sett kurven?

Because this is a yes/no question. In Norwegian main-clause yes/no questions, the finite verb comes first.

• Statement: Du har ikke sett kurven.
• Question: Har du ikke sett kurven?

So har moves to the front, just like English often does with have in Have you...?

This is the present perfect tense.

• har = the auxiliary verb have
• sett = the past participle of å se = to see

So har sett means have seen.

The sentence is asking whether the person has seen the basket up to now, not just at one specific finished moment in the past.

Sett is the past participle of the irregular verb å se.

Some forms of å se are:

• infinitive: å se
• present: ser
• past: så
• past participle: sett

So:

• Jeg ser = I see
• Jeg så = I saw
• Jeg har sett = I have seen

In Norwegian, ikke usually comes after the subject in a main clause, but before the main verb or participle if there is an auxiliary.

Here the structure is:

• Har = finite auxiliary
• du = subject
• ikke = negation
• sett = participle

So Har du ikke sett ...? is the normal order.

Compare:

• Du har ikke sett kurven. = statement
• Har du ikke sett kurven? = question

Yes, often it does. Har du ikke sett kurven? can suggest:

• surprise
• mild impatience
• an expectation that the person probably has seen it

It often feels like English Haven’t you seen the basket? rather than a totally neutral Have you seen the basket?

A more neutral question would be:

• Har du sett kurven?

Because Norwegian usually puts the definite article at the end of the noun.

• en kurv = a basket
• kurven = the basket

So -en is doing the job that the does in English.

That means kurven already includes the idea of the.

Yes, you could. That uses the simple past instead of the present perfect.

• Har du ikke sett kurven? = Have you not seen the basket?
• Så du ikke kurven? = Didn’t you see the basket?

The difference is roughly:

• har sett connects the action to the present
• så points more directly to a finished past moment

In everyday Norwegian, the choice can sometimes overlap more than in English, but the distinction still exists.

It would be:

Du har ikke sett kurven.

That means You have not seen the basket.

This is a useful comparison because it shows the normal change from statement to yes/no question:

• statement: Du har ikke sett kurven.
• question: Har du ikke sett kurven?

The main change is that the finite verb har moves to the front.

Yes. Du means you when speaking to one person.

If you were talking to several people, you would use dere:

• Har dere ikke sett kurven? = Haven’t you all seen the basket?

So the original sentence is addressed to one person.