Kortudstederen kan spærre kortet, forudsat at jeg giver dem kortnummeret.

Because Danish normally writes compound nouns as a single word.

Here, kortudstederen is built like this:

• kort = card
• udsteder = issuer
• kortudsteder = card issuer
• kortudstederen = the card issuer

This is very common in Danish. English often uses several words where Danish uses one compound noun.

Those are the usual definite singular endings in Danish.

• -en is common with common gender nouns
• -et is common with neuter nouns

In this sentence:

• udsteder → udstederen = the issuer
• kort → kortet = the card
• kortnummer → kortnummeret = the card number

So Danish often adds the equivalent of the to the end of the noun instead of putting it in front.

Because in Danish, the definite article is usually attached to the noun itself.

So:

• kortet = the card
• kortnummeret = the card number

You do get separate words like den or det, but usually when there is an adjective or extra emphasis, for example:

• det nye kort = the new card

Without an adjective, Danish usually prefers the ending on the noun.

For the same reason: Danish compounds are usually written as one word.

• kort = card
• nummer = number
• kortnummer = card number
• kortnummeret = the card number

A useful rule: in a Danish compound noun, the last part determines the grammatical gender and the basic type of thing it is. Since nummer is neuter, kortnummer is also neuter, which is why the definite form is kortnummeret.

Because after a modal verb like kan, Danish uses the bare infinitive.

So:

• kan spærre = can block

Not:

• kan spærrer

Compare:

• Kortudstederen kan spærre kortet = The card issuer can block the card.
• Kortudstederen spærrer kortet = The card issuer blocks the card.

In the first sentence, kan is the finite verb, so spærre stays in the infinitive.

It can suggest more than one of those, and in an administrative sentence like this, it often means something like:

• is able to
• is allowed to
• has the authority to

So here kan is not just physical ability. It sounds more like the issuer may/can block the card under the stated condition.

This is very common in rules, policies, and formal statements.

In this context, spærre means to block, to freeze, or to deactivate a card so it cannot be used.

A very common expression is:

• spærre et kort = block a card

It does not mean physically locking something up. It means stopping the card from functioning.

Forudsat at means provided that, assuming that, or on condition that.

Yes, it is fairly formal and is especially common in:

• official writing
• contracts
• banking language
• rules and conditions

A less formal alternative would often be hvis = if, but forudsat at sounds more precise and legal/administrative.

Because forudsat at introduces a subordinate clause.

In Danish:

• main clauses usually follow the V2 rule, where the finite verb comes in second position
• subordinate clauses usually have subject + verb order

So:

• Main clause: Kortudstederen kan spærre kortet
• Subordinate clause: at jeg giver dem kortnummeret

That is why you get jeg giver, not giver jeg.

Because dem is the object form, while de is the subject form.

Here, them is receiving something, so Danish uses dem:

• jeg giver dem kortnummeret = I give them the card number

Compare:

• De spærrer kortet = They block the card
• Jeg giver dem kortnummeret = I give them the card number

Also, capitalized De/Dem can be a formal way of saying you, but here the lowercase dem almost certainly means them.

Because kortudstederen refers to an institution or company, and Danish can naturally switch to plural dem when talking about the people behind that institution.

So the idea is:

• kortudstederen = the issuer as an organization
• dem = the people at the issuer / the company in practice

English does something similar:

• The bank can block the card if I give them the card number.

So even though the noun is singular, the plural pronoun can still sound natural.

Both are possible, but giver dem kortnummeret is the more compact and natural version here.

Danish often allows two patterns:

• jeg giver dem kortnummeret
• jeg giver kortnummeret til dem

The first is a double-object pattern and is very common, especially when the recipient is a pronoun like dem.

The second version with til is also grammatical, but it can sound a bit heavier or more emphatic.

Because forudsat at jeg giver dem kortnummeret is a subordinate clause, and many Danish writers put a comma before subordinate clauses.

This depends on the comma system being used:

• with startkomma, the comma is written
• with nyt komma, it may be left out

So both of these can be acceptable:

• Kortudstederen kan spærre kortet, forudsat at jeg giver dem kortnummeret.
• Kortudstederen kan spærre kortet forudsat at jeg giver dem kortnummeret.

The version with the comma is very normal.