Chapter 4.
The Holy Spirit is one and the same Who spoke in the prophets and apostles, Who is the Spirit of God and of Christ; Whom, further, Scripture designates the Paraclete, and the Spirit of life and truth.
55. But no one will doubt that the Spirit is one, although very many have doubted whether God be one. For many heretics have said that the God of the Old Testament is one, and the God of the New Testament is another. But as the Father is one Who both spoke of old, as we read, to the fathers by the prophets, and to us in the last days by His Son; and as the Son is one, Who according to the tenour of the Old Testament was offended by Adam, seen by Abraham, worshipped by Jacob; so, too, the Holy Spirit is one, who energized in the prophets, was breathed upon the apostles, and was joined to the Father and the Son in the sacrament of baptism. For David says of Him: “And take not Your Holy Spirit from me.” And in another place he said of Him: “Whither shall I go from Your Spirit?”
56. That you may know that the Spirit of God is the same as the Holy Spirit, as we read also in the Apostle: “No one speaking in the Spirit of God says Anathema to Jesus and no one can say, Lord Jesus, but in the Holy Spirit,” the Apostle calls Him the Spirit of God. He called Him also the Spirit of Christ, as you read: “But you are not in the flesh but in the Spirit, if so be that the Spirit of God dwells in you.” And farther on: “But if the Spirit of Him Who raised Jesus from the dead dwells in you.” The same is, then, the Spirit of God, Who is the Spirit of Christ.
57. The same is also the Spirit of Life, as the Apostle says: “For the law of the Spirit of Life in Christ Jesus has delivered me from the law of sin and death.”
58. Him, then, Whom the Apostle called the Spirit of Life, the Lord in the Gospel named the Paraclete, and the Spirit of Truth, as you find: “And I will ask the Father, and He will give you another Comforter [Paraclete], that He may be with you for ever, even the Spirit of Truth, Whom this world cannot receive; because it sees Him not, neither knows Him.” You have, then, the Paraclete Spirit, called also the Spirit of Truth, and the invisible Spirit. How, then, do some think that the Son is visible in His Divine Nature, when the world cannot see even the Spirit?
59. Receive now the saying of the Lord, that the same is the Holy Spirit Who is the Spirit of Truth, for you read in the end of this book: “Receive the Holy Spirit.” And Peter teaches that the same is the Holy Spirit Who is the Spirit of the Lord, when he says: “Ananias, why has it seemed good to you to tempt and to lie to the Holy Spirit?” And immediately after he says again to the wife of Ananias: “Why has it seemed good to you to tempt the Spirit of the Lord?” When he says “to you,” he shows that he is speaking of the same Spirit of Whom he had spoken to Ananias. He Himself is, then, the Spirit of the Lord Who is the Holy Spirit.
60. And the Lord Himself made clear that the same Who is the Spirit of the Father is the Holy Spirit, when according to Matthew He said that we ought not to take thought in persecution what we should say: “For it is not you that speak, but the Spirit of your Father that speaks in you.” Again He says according to St. Luke: “Be not anxious how you shall answer or speak, for the Holy Spirit of God shall teach you in that hour what you ought to say.” So, although many are called spirits, as it is said: “Who makes His Angels spirits,” yet the Spirit of God is but one.
61. Both apostles and prophets received that one Spirit, as the vessel of election, the Doctor of the Gentiles, says: “For we have all drunk of one Spirit;” Him, as it were, Who cannot be divided, but is poured into souls, and flows into the senses, that He may quench the burning of this world's thirst.
Source: On the Holy Spirit (New Advent)