The Holy Spirit is Life equally with the Father and the Son, in truth whether the Father be mentioned, with Whom is the Fount of Life, or the Son, that Fount can be none other than the Holy Spirit.
171. We have said that the Father is Light, the Son is Light, and the Holy Spirit is Light; let us also learn that the Father is Life, the Son Life, and the Holy Spirit Life. For John said: “That which was from the beginning, that which we have heard, and which we have seen, and have beheld with our eyes, and our hands have handled concerning the Word of Life; and the Life appeared, and we saw and testify, and declare to you of that Life which was with the Father.” He said both Word of Life and Life that he might signify both the Father and the Son to be Life. For what is the Word of Life but the Word of God? And by this phrase both God and the Word of God are shown to be Life. And as it is said the Word of Life, so, too, the Spirit of Life. Therefore, as the Word of Life is Life, so, too, the Spirit of Life is Life.
172. Learn now that as the Father is the Fount of Life, so, too, many have stated that the Son is signified as the Fount of Life; so that, he says, with You, Almighty God, Your Son is the Fount of Life. That is the Fount of the Holy Spirit, for the Spirit is Life, as the Lord says: “The words which I speak unto you are Spirit and Life,” for where the Spirit is, there also is Life; and where Life is, is also the Holy Spirit.
173. Many, however, consider that in this passage the Father only is signified by the Fount. Let them, however, notice what the Scripture relates: “With You is the Well of Life.” That is, the Son is with the Father; since the Word was with God, Who was in the beginning, and was with God.
174. But whether in this place one understands the Fount to be the Father or the Son, we certainly do not understand a fount of that water which is created, but the Fount of that divine grace, that is, of the Holy Spirit, for He is the living water. Wherefore the Lord said: “If you know the gift of God, and Who He is that says to you, Give me to drink, you would have asked Him, and He would have given you living water.”
175. This was the water for which the soul of David thirsted. The hart desires the fountain of these waters, not thirsting for the poison of serpents. For the water of the grace of the Spirit is living, that it may purify the inner parts of the mind, and wash away every sin of the soul, and purify the transgression of hidden faults.
Source: On the Holy Spirit (New Advent)