Yes, you are granted a spellbook by your spellcasting.
Spellcasting (PHB 114):
As a student of arcane magic, you have a spellbook containing spells
that show the first glimmerings of your true power.
So you have a spellbook, disregarding how you got it.
Class Features (PHB 164):
When you gain a new level in a class, you get its features for that
Spellcasting is a class feature that has additional rules tied to it.
Spellcasting (PHB 164):
Your capacity for spellcasting depends partly on your combined levels
in all your spellcasting classes and partly on your individual levels
in those classes.
As the additional rules do not state that you lose any spellbook granted, you have one.
So what about the spellbook that is listed under equipment?
Well, that is also a separate issue (which most GM's dutifully ignore).
Your spellbook is part of the spellcasting class feature, which is a separate instance from your starting equipment. Your starting equipment is not a class feature granted by multiclassing PHB errata V2.0 page 2
You don’t, however, receive the class’s starting equipment,
So what is your new spellbook that you get due to your spellcasting feature?
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.
Well, it could be lots of things, Rubiksmoose phrases it nicely:
you can use almost anything as a spellbook even things like metal
sheets and carved rocks in a bag.