Who or What is the Holy Spirit?
The Godhead (essential deity) exists in what we call a tri-unity—three identical persons, but one substance. The Father, the Son and the Holy Spirit. (see all three mentioned in Matthew 28.19 and 2 Corinthians 13.14)
Now, there are not three Gods, but one God in one substance, but that one God exists in three identical persons differentiated by three economic functions.
It is not the case that the one God simply manifests Himself in three different ways at different times. No, each person is present in the other. And the Spirit (of God) or Holy Spirit proceeds from and executes the will of the Father and the Son and glorifies the latter on earth. The Holy Spirit as the third member of the trinity, but the ‘clone’ of the Son.
Hence, with the presence of the Spirit in the believer, it may truly be said that Christ or the Spirit of Christ resides in us. All that said, one must humbly hasten to add, that, while this teaching of the Scriptures is more fully disclosed in the NT, this trinitarian existence of the Godhead is a great mystery that transcends our ability to get our heads completely around. No exact analogy to it exists, though one can come up with analogies that may approach it. Yet all, because imperfect, are latent with potential misleading errors.
The ideal of the trinitarian nature of the Godhead, I believe, projects itself into God’s created order. For in the Trinity there is a perfect unity complemented by majestic diversity combined with a blessed community and perfect harmony.