Я бачу багато знайомих людей у спортзалі.

Both come from the same noun, but they are different cases:

• люди – nominative plural (“people” as the subject)
• людей – genitive plural (“of people”, “people” after certain words)

In this sentence we have багато людей.
The word багато (“many / a lot of”) always requires the genitive case after it for countable nouns.
So you must say:

• ✅ багато людей – many people
• ❌ багато люди – incorrect

That’s why it’s людей, not люди.

знайомий is an adjective: “familiar / known”. It must agree with the noun людей in:

• number (plural)
• case (genitive)

So the forms are:

• nominative plural: знайомі люди – “familiar people” (as a subject)
• genitive plural: знайомих людей – “(of) familiar people” (after багато)

Since людей is genitive plural, the adjective must also be genitive plural:

• ✅ багато знайомих людей
• ❌ багато знайомі люди

In Ukrainian, many quantity words are followed by the genitive case.
багато (“many, a lot of”) is one of them.

• багато людей – many people
• багато води – a lot of water
  
  • “людей” – genitive plural (countable)
  • “води” – genitive singular (uncountable)

So the rule is: after багато, put the noun (and its adjectives) in the genitive.

Yes. That is also correct, but the nuance changes slightly.

• багато знайомих людей – literally “many familiar people” (emphasis on “people that are familiar to me”)
• багато знайомих – “many acquaintances / many people I know”

In Ukrainian, знайомий can be:

• an adjective: знайомі люди – familiar people
• a noun: знайомі – acquaintances

So both sentences are fine:

• Я бачу багато знайомих людей у спортзалі. – I see many familiar people at the gym.
• Я бачу багато знайомих у спортзалі. – I see many acquaintances at the gym.

Yes, в спортзалі is also grammatically correct.

Ukrainian has two variants of the same preposition: в and у.
They are chosen mainly for euphony, to make pronunciation easier and smoother.

• After a word ending in a consonant, speakers often prefer у:
  людей у спортзалі (й + у sounds smooth)
• After a word ending in a vowel, в is often more natural:
  я в спортзалі

So:

• у спортзалі and в спортзалі both mean “in the gym”.
• In this exact sentence, …людей у спортзалі sounds slightly smoother than …людей в спортзалі, but both are acceptable.

In Ukrainian:

• у / в
  • locative is used for being inside something:
    
    • у спортзалі – in the gym
    • у кімнаті – in the room
• на is used for surfaces and many set expressions:
  
  • на столі – on the table
  • на роботі – at work
  • на стадіоні – at the stadium

A gym is treated as an enclosed space you are inside, so you say у спортзалі, not на спортзалі.

The base word is спортзал (“gym”). The form спортзалі is:

• locative case, singular (also called “prepositional” in some books)
• used mostly after в / у, на, по to show location

Declension (singular):

• nominative: спортзал – the gym (subject)
• locative: у спортзалі – in the gym

The change -ал → -алі is regular for many masculine nouns in the locative:

• зал → у залі – in the hall
• спортзал → у спортзалі – in the gym

бачу is:

• present tense
• 1st person singular (“I”)
• from the imperfective verb бачити (“to see”)

Ukrainian has only one present tense for imperfective verbs. It covers both English:

• “I see”
• “I am seeing”

So Я бачу багато знайомих людей у спортзалі can be translated as:

• “I see many familiar people at the gym.”
• “I’m seeing many familiar people at the gym.”

English chooses the form based on context; Ukrainian uses бачу in both cases.

Yes, that word order is natural and correct. Ukrainian word order is flexible, and changes of order mostly affect emphasis, not grammar.

Some possible versions:

1. Я бачу багато знайомих людей у спортзалі.
   – neutral, focus on what you see.

2. У спортзалі я бачу багато знайомих людей.
   – slight emphasis on “at the gym”.

3. Багато знайомих людей я бачу у спортзалі.
   – strong emphasis on “many familiar people”.

All are grammatically fine; you just choose where you want the emphasis.

Ukrainian has no articles (no equivalent of English a / an / the).
The sentence Я бачу багато знайомих людей у спортзалі can mean:

• “I see many familiar people at the gym.”
• “I see many familiar people at a gym.”

Whether it’s “the” or “a” is understood from context, not from any special word. Speakers rely on:

• context
• word order and stress
• sometimes demonstratives like цей / той (“this / that”) if they need to be extra clear

Literally, знайомий = “familiar / known”.

In this phrase:

• знайомих людей = “people who are familiar to me, people I know”

Natural English translations:

• “familiar people”
• “people I know”

If you want to say specifically “acquaintances” (as a noun), Ukrainian often just uses plural знайомі:

• Я бачу багато знайомих у спортзалі. – I see many acquaintances at the gym.

So:

• знайомих людей – emphasizes them as people, adds a slightly descriptive tone.
• знайомих – sounds more like “acquaintances”.

You just add не (not) before the verb:

• Я не бачу багато знайомих людей у спортзалі.

Literally: “I not see many familiar people in the gym.”
Word-for-word: Я (I) не бачу (do not see) багато знайомих людей (many familiar people) у спортзалі (at the gym).

Pronunciation (simplified): [sport-ZA-lee]

• спортзалі – stress on за: спортза́лі
• syllables: спорт-за-лі

Syllable stress in the full sentence:

• Я́ ба́чу бага́то знайо́мих людéй у спортза́лі.

You don’t need to write the stress marks in normal writing; they are just for learners.

Yes, людей is the genitive plural of люди (“people”).
There is no singular form люда in normal modern Ukrainian.

For one person, Ukrainian uses a different noun: людина – “a person, human being”.

Examples:

• одна людина – one person
• дві людини – two people
• багато людей – many people

So in your sentence, людей clearly means more than one person.