Je tu hodně prostoru.

Czech does not need a dummy subject like English there in There is a lot of space here.

So the sentence is built more like:

• Je = is
• tu = here
• hodně prostoru = a lot of space

Literally, it is something like Is here a lot of space, which is normal in Czech. The idea of English there is is expressed just by the verb plus the rest of the sentence.

Tu means here.

It is very close in meaning to:

• tady = here
• zde = here, but more formal or written

In everyday Czech, tu and tady are both common. In this sentence, you could also say:

• Je tady hodně prostoru.
• Je zde hodně prostoru.

All are correct, but zde sounds more formal.

Because hodně usually requires the genitive form of the noun.

The basic noun is:

• prostor = space, room, area

After hodně (a lot of), it changes to:

• prostoru = of space

So:

• hodně prostoru = a lot of space

This is very common in Czech:

• hodně času = a lot of time
• hodně vody = a lot of water
• hodně lidí = a lot of people

Here it is genitive singular.

That may feel surprising to an English speaker, because English space does not show this kind of change. But in Czech, prostor is being treated as a noun meaning space/room, and after hodně it appears in the genitive singular:

• nominative: prostor
• genitive singular: prostoru

So hodně prostoru is literally something like a lot of space or much space, not many spaces.

Because Czech commonly uses singular je in this kind of existential sentence with an amount expression like hodně prostoru.

So even though hodně suggests a large quantity, the whole expression behaves like a single amount:

• Je tu hodně prostoru. = There is a lot of space here.

This singular verb is very normal in Czech with this structure.

Here prostor means space or room, especially in the sense of available physical space.

It overlaps with místo, but they are not always identical:

• prostor = space, room, area, capacity
• místo = place, spot, seat, room/space

In this sentence, prostor sounds very natural for there is plenty of space.

You could also hear:

• Je tu hodně místa.

That also means There is a lot of room here and is extremely common. In many everyday situations, hodně místa may even sound more natural than hodně prostoru, depending on context.

Yes. Czech word order is flexible, and the meaning stays basically the same.

Possible versions include:

• Je tu hodně prostoru.
• Je tady hodně prostoru.
• Hodně prostoru tu je.
• Tu je hodně prostoru.

The differences are mostly about emphasis or style, not basic meaning.

The original Je tu hodně prostoru is a very neutral, natural way to say it.

Yes, often it would.

• hodně prostoru = a lot of space
• moc prostoru = a lot of space / lots of space

In everyday speech, moc is very common and often sounds a bit more conversational.

So you can say:

• Je tu hodně prostoru.
• Je tu moc prostoru.

Both are natural.

A rough guide is:

• hodně ≈ HOD-nye

The tricky part is dně, where the n is softened. English does not have an exact equivalent, so it may take practice.

Also remember that Czech stress is usually on the first syllable:

• HOD-ně
• PROS-to-ru

It can be both, depending on context.

Most commonly, Je tu hodně prostoru means physical room, for example in a house, office, or car.

But prostor can also be more abstract, such as:

• room for discussion
• room for improvement
• space for creativity

So in the right context, the sentence could also mean something like There is a lot of room here in a more figurative sense.