The Holy Spirit does not compete with God as another god, neither is the Holy Spirit synonyms of God.

Anything that belongs to God is considered holy. Therefore, God's own Spirit, is called the Holy Spirit of God.

When I am holding my child with my hands, you could also say that I am holding my child, or that my hands are holding the child. Both statements are correct.

The same way one could say that when the Holy Spirit touches someone, it means God is touching the person and when one touches the Holy Spirit, one is touching God.

Both was present at the creation

In the beginning, God created the heavens and the earth. The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. -- Genesis 1:1-2 (ESV)

The reasoning is that the Holy Spirit is uncreated, therefore He should be God.

He gives life

The Spirit of God has made me, and the Breath (or Spirit) of the Almighty gives me life. -- Job 33:4 (ESV)

Job does not contradict, but complements Paul:

It is written,

“I have made you the father of many nations” (Genesis 17:5)

in the presence of the God in whom he believed, who gives life to the dead and calls into existence the things that do not exist.

-- Romans 4:17

He made the Son of God's mother pregnant

When his mother Mary had been betrothed to Joseph, before they came together she was found to be with child from the Holy Spirit. -- Matthew 1:18 (ESV)

and

And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God. -- Luke 1:35

If the Son of God (Matthew 16:16)'s father is the Holy Spirit, then the Holy Spirit must be God the Father.

He represents God

But Peter said, “Ananias, why has Satan filled your heart to lie to the Holy Spirit and to keep back for yourself part of the proceeds of the land?
While it remained unsold, did it not remain your own?
And after it was sold, was it not at your disposal?
Why is it that you have contrived this deed in your heart?
You have not lied to man but to God.”

-- Acts 5:3-4

And Paul quotes the Holy Spirit's words:

Therefore, as the Holy Spirit says,

“Today, if you hear his voice, do not harden your hearts as in the rebellion, on the day of testing in the wilderness, where your fathers put me to the test and saw my works for forty years. Therefore I was provoked with that generation, and said, ‘They always go astray in their heart; they have not known my ways.’ As I swore in my wrath, ‘They shall not enter my rest.’” (Psalm 95:7-11)

Take care, brothers, lest there be in any of you an evil, unbelieving heart, leading you to fall away from the living God.

-- Hebrews 3:7-12 (ESV)

He searches God

But, as it is written,

“What no eye has seen, nor ear heard, nor the heart of man imagined, what God has prepared for those who love him” (Isaiah 64:4)

these things God has revealed to us through the Spirit. For the Spirit searches everything, even the depths of God. For who knows a person’s thoughts except the spirit of that person, which is in him? So also no one comprehends the thoughts of God except the Spirit of God. Now we have received not the spirit of the world, but the Spirit who is from God, that we might understand the things freely given us by God. And we impart this tin words not taught by human wisdom but taught by the Spirit, interpreting spiritual truths to those who are spiritual. The natural person does not accept the things of the Spirit of God, for they are folly to him, and he is not able to understand them because they are spiritually discerned. -- 1 Corinthians 2:9-14 (ESV)

He is eternal

For if the blood of goats and bulls, and the sprinkling of defiled persons with the ashes of a heifer, sanctify for the purification of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to serve the living God. -- Hebrews 9:11 (ESV)

He is omnipresent

Where shall I go from your Spirit?

Or where shall I flee from your presence?
If I ascend to heaven, you are there!
If I make my bed in Sheol, you are there!
If I take the wings of the morning dwell in the uttermost parts of the sea, even there your hand shall lead me, and your right hand shall hold me.
If I say, “Surely the darkness shall cover me, and the light about me be night,” even the darkness is not dark to you; the night is bright as the day, for darkness is as light with you.

-- Psalm 139:7-12 (ESV)

He is good

Jesus said:

"Why do you call me good? No one is good except God alone." -- Mark 10:18 (ESV)

The David wrote:

Let your good Spirit lead me on level ground! -- Psalm 143:10 (ESV)

Nehemiah wrote:

You gave your good Spirit to instruct them... -- Nehemiah 9:20