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Apologies in advance if this is a duplicate, I could (to my surprise) not find a question on this (there is only one for Pathfinder).
The question is simple: is a normal spellbook in 5e a magical item or not? (Not asking about the Grimoire from TCE, nor about the Awakened Spellbook from an Order of Scribes wizard).
A spellbook is described in the adventuring gear section of the PHB thusly:
Essential for wizards, a spellbook is a leather-bound tome with 100 blank vellum pages suitable for recording spells.
Page 114 of the PHB in the wizard section describes it as follows:
The Book’s Appearance. Your spellbook is a unique compilation of spells, with its own decorative flourishes and margin notes. It might be a plain, functional leather volume that you received as a gift from your master, a finely bound gilt-edged tome you found in an ancient library, or even a loose collection of notes scrounged together after you lost your previous spellbook in a mishap.
Spellbook is not listed in the magic item section of the DMG
Spellbook is in the list of adventuring gear
As a counterexample, the potion of healing, a common magic item that appears on the adventuring gear list, also appears in the magic item list, and its description there explicitly states it is a "magical red fluid". (Thanks to commenters). The potion is the only individual item in the table that is in italics, a convention often used for magic items. The spellbook is in plain font type.
The spellbook is just a book (or other substrate for your unique system of recording spells) of high quality, that you use expensive, special inks to write spells into.
This inofficial Jeremy Crawford tweet confirms this
A normal spellbook or spellcasting focus is not a magic item.
Would inscribing a spell into the book maybe suffice to turn it from a mundane item into a magic item? This would be against the rules on how to create magic items, which require crafting (page 128, DMG). There is no specific rule for turning mundane into magical spellbooks in the core rules to override this.
Because a normal spellbook is not a magic item, it does not radiate magic and cannot be detected with detect magic. It also does not benefit from the increased resilience to damage that magic items enjoy (such as resistance to all damage types), and needs to be protected appropriately from harm.
Just like there are normal swords and magic swords, there can be spellbooks that are magic items. Thasha's Cauldron of Everything lists several of them, for example the Arcane Grimoire, Astromancy Archive or Atlas of Endless Horizons. Just the normal spellbook is not a magic item.