Is the Holy Spirit, the Spirit of the Father? It's a pretty simple question, so I don't really know if there is much I can add to the body.
I will answer from the Trinitarian perspective.
The two terms are synonymous...
...and yet, while the Father "is spirit"...
...the Father is not "the Holy Spirit."
If the Holy Spirit was just the Father in His spirit form, then He would in fact be speaking on His own initiative. Since Jesus makes it clear that the Spirit will not speak on His own initiative, but will do what He is told (basically), that indicates there is One with more authority who is "over" Him.
Conclusion: From these passages we can conclude that "the Holy Spirit" is not the Father (though the Father "is spirit") and yet, the "Spirit of your Father" is synonymous with "the Holy Spirit." Perhaps the best explanation is that the Spirit is "of" the Father in the sense that He comes forth from the Father, without confusing the two as if they were one Person.
(Although I'm willing to answer my own question, no one should let that deter them from also submitting an answer. Also, for the record, I'm a Trinitarian.)
Yes, the Holy Spirit is the Spirit of the Father. This is proven by many scriptures.
If we examine the following synoptic parallelism, Luke and Mark identify the one speaking in the apostles as the "Holy Spirit," but Matthew identifies it as "the Spirit of your Father" (i.e., "your Father's Spirit"). Therefore, the Holy Spirit is the Spirit of the Father.
In De Trinitate ("On the Trinity"), Book XV, Ch. XXVI, St. Augustine wrote,
which is translated as,
As Augustine notes, the Holy Spirit is the Spirit of both the Son and the Father. Other scriptures in the New Testament also prove this.
The Holy Spirit is the Spirit of the Father:
The phrase "of Him who raised Jesus from the dead" refers to the Father. "The Spirit of Him" dwells in the Christian. "He who raised Christ from the dead" is the Father. "His Spirit" is the Father's Spirit, the Holy Spirit, who dwells in the Christian.
The Holy Spirit is frequently referred to as "His Spirit," where the possessive pronoun "His" evidently refers to the antecedent "God the Father" (cp. Num. 11:29; Isa. 48:16; 1 Cor. 2:10; Eph. 3:16; 1 John 4:3). Therefore, the Holy Spirit is the Father's Spirit.
The Holy Spirit is the Spirit of the Son:
If Adonay is referring to the Father, yes.
I personally have always pictured the trinity in a particular way, that helps me understand it. I don't believe there's anything biblical to say 'this is how it is' but I don't think it contradicts anything that the Bible says either - if it does I'd appreciate someone pointint that out.
I generally see:
They are all God, like you consist of multiple parts (body, mind, soul), it is difficult to reconcile these things together but combined they all describe who you are. The same is true for God, all these combined represent who God is.
I'm going to go out on a limb and say "No", the Holy Spirit is not the Spirit of the Father. That would imply the Father is something comprised of Spirit and something(s) else, or that the Holy Spirit is lesser than the Father or the Son. The general idea is that all three persons of the trinity are co-equal and co-eternal. The Bible is pretty clear that God is Spirit. That means all the persons of God are Spirit.
It doesn't help that the basic creeds describe the Son as begotten by the Father, or the Holy Spirit as proceeding from the Father and from/through the Son, as those terms imply causation and temporal ordering.
The way I think about it currently is there is one uncreated nature which is God (I find using terms other than "being" or "person" is helpful to disambiguate what we're talking about). The Father personifies the Will or Way of God, the Son personifies the Reason or Truth of God, and the Holy Spirit personifies the Life of God.
Is the Holy Spirit, the Spirit of the Father?
Yes, That is the part of God that carries out the word of God, or the will of God. God speaks his word and the Holy Spirit makes sure it get don't according to God's will. Genesis 1;2 The Spirit of God was moving over the surface of the water, waiting for God to speak.
Even today the Holy Spirit is waiting for you to speak God's word, It is hard to be effective witness with out it, in you to carry out God's will. Acts 1; 4,5 this tells you that you need the baptism of the Holy Spirit or you witness will be weak. John baptized with water you will be baptized with the Holy Spirit and fire. The world is still happening and it still needs witness, to be a effective witness you need it. We didn't walk with Jesus we didn't see Jesus were not call to be apostle, No but you are called to be a witness. Not a run down and kick in the head witness, But one who has the power of God following out of them to destroy the works of the enemy on the Earth. Read Mark chapter 4 again Don't Kick people or hit them over the head with it either hit yourself over the head with it. Then learn to love people with it. It works.