tosyokan ha sizuka da.

は is the topic marker. It tells us that “図書館” (the library) is what we’re talking about. In English you might think of it like saying “As for the library, (it is…)”.

• は sets or contrasts the topic.
• が marks a new subject or emphasizes it (“There is/ it is …”).

If you said 図書館が静かだ, you’d be focusing on the fact that “it is the library that is quiet,” perhaps in contrast to something else.

だ is the plain (dictionary) form of the copula, similar to the English verb “to be” (is/am/are).

• After な-adjectives (like 静か), you attach だ to make a complete predicate.
• Without だ, it would be incomplete in plain speech:
  
  • 図書館は静か sounds unfinished (more like an adjective stuck to the topic).

In polite speech you’d use です instead of だ (図書館は静かです).

When using 静か as an adjective before a noun, you must add な (静かな図書館). But when 静か is the predicate (describing the topic after the verb “to be”), you drop the な and use the copula:

• Before noun: 静か
  • な
    • noun → 静かな図書館
• After topic: 静か
  • だ → 図書館は静かだ

Japanese follows a topic–comment (or subject–predicate) order:

1. Topic/Subject → 2. Predicate (description/verb)
   Here:
2. 図書館は (topic)
3. 静かだ (predicate)
   Switching them would break that basic pattern and sound ungrammatical.

In very casual spoken Japanese, you can drop だ after な-adjectives and nouns:

• Casual: 図書館は静か (sounds relaxed/informal)
• Slightly more formal/plain: 図書館は静かだ
• Polite: 図書館は静かです

Dropping だ is common among friends or in diary-style writing.

It’s a matter of politeness level:

• 静かだ → plain, informal (used in casual conversation among peers or in writing like novels)
• 静かです → polite, formal (used with strangers, elders, in business settings)

Both mean “is quiet” but fit different social situations.