The sefer Shoshan Sodos5 says that in the soul of Adam, the initial thought of Hashem that Adam will live forever was still there and also that which He said ביום אכלך ממנו מות תמות, on the day you eat from the eitz hadaas you will surely die.6 How can these two coexist?
Adam would live 1,000 years which is a day by Hashem as it says כי אלף שנים בעיניך 1,000 years are like a day. Therefore this fulfills "on the day you eat from the eitz hadaas you will surely die," as according to this, he died on the day he ate from the eitz hadaas. That is a day in Hashem's eyes, not in what we call a day.
We know that Adam gave 70 of his 1,000 years to Dovid and therefore he only lived until 930.9 When Hashem said מות תמות a deeper explanation is that he will die twice and will not live 1,000 years uninterrupted. That is to say, Adam will die at 930 and then he will continue through Dovid who will die at 70. So מות תמות refers to the death of Adam and Dovid. In the future that same Neshama of Adam that continued with Dovid will then be in Moshiach and that will fulfill Hashem's initial thought of Adam living forever.
These three are hinted at in the name אדם, דוד, משיח which stands for Adam, Dovid, Moshiach.
5 Sod ki byom.
6 Breishis 2:17.
7 Tehillim 90:4.
8 See Bamidbar Rabba 14:12, Zohar Vayishlach 168a, Pirkei Drebbe Eliezer, Chapter 19.
9 We find a hint that Dovid rectified Adam with his 70 years. It says והמלך דוד זקן בא בימים (Melachim Alef, 1:1). The Chida (Chomas Anach, Melachim Alef, 1) points out that the end letters of דוד בא בימים spells אדם while the final letters of והמלך זקן have a gematria of 70, the number of years Dovid lived, rectifying Adam.