St Augustine of AfricaThe Apostle Paul says: To each of us is given grace according to the measure of the donation of Christ.

To show that the donation of Christ is the Holy Spirit he went on to add, That is why it says, he ascended on high, he took captivity captive, he gave gifts to men.

But it is public knowledge that when the Lord Jesus had ascended to heaven after his resurrec­tion from the dead he gave the Holy Spirit.

[…] And do not let it worry you that he says ‘gifts’, not ‘gift’. He was quoting the text from a psalm: you have ascended on high, you have taken captivity captive, you have received gifts among men.

Elsewhere he mentions many gifts, and then says, All these does one and the same Spirit achieve, distributing them severally to each as he wills.

The same word is found in the Letter to the Hebrews, where it is written, God bearing witness with signs and portents and various mighty deeds and distribu­tion of the Holy Spirit. […] The Apostle Peter, speaking about Christ to the Jews who were moved in their hearts and saying, What shall we do, brothers? said to them:

Repent, and let each one of you be baptised in the name of Jesus Christ for the forgiveness of sins, and you shall receive the gift of the Holy Spirit.

And there is a lot more scriptural evidence which all conspires to prove that the Holy Spirit is the gift of God, in that he is given to those who love God through him.

[…] Insofar as people now see that the Holy Spirit is called the gift of God, they must realize of course that when they hear the phrase, the gift of the Holy Spirit, they are to recognize the same figure of speech as in the phrase, in the stripping of the body of flesh.

Just as the body of flesh is nothing but flesh, so the gift of the Holy Spirit is nothing but the Holy Spirit.

So he is the gift of God insofar as he is given to those to whom he is given; but in himself he is God even if he is not given to anyone, because he was God, co-eternal with the Father and the Son, even before he was given to anyone.

Nor is he less than the Father and the Son because they give and he is given: he is given as God’s gift in such a way that as God he also gives himself.

Augustine of Hippo (354-430:  On the Trinity 15,  from the Monastic Office of Vigils, Wednesday of the Seventh Week of Ordinary Time, Year 1.