Is the Holy Spirit, the Spirit of the Father? 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...

But when they hand you over, do not worry about how or what you are to say; for it will be given you in that hour what you are to say. For it is not you who speak, but it is the Spirit of your Father who speaks in you. -Matthew 10:19-20
When they arrest you and hand you over, do not worry beforehand about what you are to say, but say whatever is given you in that hour; for it is not you who speak, but it is the Holy Spirit. -Mark 13:11

...and yet, while the Father &quot;is spirit&quot;...

God is spirit, and those who worship Him must worship in spirit and truth. -John 4:24

...the Father is not &quot;the Holy Spirit.&quot;

But when He, the Spirit of truth, comes, He will guide you into all the truth; for He will not speak on His own initiative, but whatever He hears, He will speak; and He will disclose to you what is to come. -John 16:13

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 &quot;over&quot; Him.
Conclusion: From these passages we can conclude that &quot;the Holy Spirit&quot; is not the Father (though the Father &quot;is spirit&quot;) and yet, the &quot;Spirit of your Father&quot; is synonymous with &quot;the Holy Spirit.&quot; Perhaps the best explanation is that the Spirit is &quot;of&quot; the Father in the sense that He comes forth from the Father, without confusing the two as if they were one Person.