В библиотеке можно не только читать, но и писать конспект.

Because after в when you mean location (in / at a place), Russian normally uses the prepositional case.

• библиотека = library
• в библиотеке = in the library / at the library

So:

• Я в библиотеке. = I am in the library.
• Читать в библиотеке = to read in the library

If в means motion into a place, Russian usually uses the accusative instead:

• идти в библиотеку = to go to the library

So the contrast is:

• в библиотеке = in the library
• в библиотеку = into / to the library

Russian often uses impersonal sentences, where no specific subject like I, you, or people is stated.

Here, можно means something like:

• it is possible
• it is allowed
• one can

So:

• В библиотеке можно... = In the library, one can... / It is possible to... in the library

It is intentionally general. The sentence is talking about what people in general can do there, not about one specific person.

Можно is a very common Russian word that can mean:

• it is possible
• it is allowed
• you may
• one can

In this sentence, it gives a general idea of possibility / permission:

• В библиотеке можно читать. = In the library, you can read.

It is often followed by an infinitive:

• можно читать = can read
• можно писать = can write

A useful contrast:

• можно = allowed / possible
• нельзя = not allowed / impossible

For example:

• В библиотеке можно читать. = You may read in the library.
• В библиотеке нельзя шуметь. = You must not make noise in the library.

Because they depend on можно.

After можно, Russian normally uses the infinitive:

• можно читать = can read
• можно писать = can write

This is similar to English can + verb, except Russian uses the infinitive form directly after можно.

So in the sentence:

• можно не только читать, но и писать конспект

both verbs stay in the infinitive because they are both governed by можно.

This is a fixed Russian pattern meaning not only ..., but also ...

So:

• не только читать, но и писать конспект = not only read, but also write/take notes

It connects two parallel ideas.

More examples:

• Он не только читает, но и пишет.
  = He not only reads, but also writes.

• Это не только интересно, но и полезно.
  = It is not only interesting, but also useful.

In your sentence, the two linked actions are:

• читать
• писать конспект

Because не только ..., но и ... links two coordinated parts of the sentence, and in Russian this structure normally takes a comma before the second part.

So:

• можно не только читать, но и писать конспект

That comma is standard punctuation in this construction.

They are imperfective because the sentence talks about general activity, not a single completed action.

• читать = to read
• писать = to write

Here the meaning is broad: what one can do in a library in general. Russian usually uses the imperfective aspect for that.

If you used perfective verbs, it would sound more like a single completed result, which does not fit as well here.

So the imperfective is natural because the sentence means something like:

• In the library, one may engage in reading and note-writing.

Конспект is a study-related word. It usually means:

• notes
• a summary
• an outline
• a set of study notes

In many contexts, писать конспект means something like:

• to take notes
• to write up notes
• to make a summary of what you read or hear

It is not exactly the same as a full formal essay. It is more like organized notes or a condensed summary.

Yes. In Russian, писать can be used in a broad sense of write, including writing down notes.

So писать конспект is understandable as:

• write a summary
• make notes
• take notes

Depending on context, Russian can also use other expressions for note-taking, but писать конспект is perfectly understandable and natural in a study context.

Because it is the direct object of писать, so it is in the accusative case.

For an inanimate masculine singular noun, the accusative looks the same as the nominative:

• nominative: конспект
• accusative: конспект

So:

• писать конспект = to write a summary / notes

If it were a different type of noun, the form might change more visibly.

Russian often uses the singular when speaking about an activity in a general way.

So писать конспект can mean:

• to write a summary
• to make notes
• to work on a set of notes

If you said писать конспекты, that would emphasize multiple summaries or multiple sets of notes.

Both can be possible depending on context, but the singular here is normal and natural.

Because Russian does not have articles like a, an, or the.

So:

• в библиотеке can mean in the library or in a library
• писать конспект can mean write a summary or sometimes write the summary, depending on context

Russian relies on context rather than articles to show whether something is definite or indefinite.

Literally, it is in the library, because в means in.

But in natural English, depending on context, at the library may also be a good translation.

So:

• В библиотеке можно читать... could be:
• In the library, you can read...
• At the library, you can read...

The Russian form itself is simply the normal way to say in the library.

It can suggest more than one of those ideas, which is very common with можно.

Depending on context, можно may mean:

• it is allowed
• it is possible
• it is appropriate / acceptable

So this sentence could imply:

• The library allows this.
• The library is a place where this can be done.
• The library is suitable not only for reading but also for making notes.

Very often, Russian leaves that nuance to context.