Men have wondered about the what the paradise mentioned in the book of Luke is. Some have said that it is its own distinct place reserved for the saints prior to the resurrection of the Lord Jesus Christ. Others believe that paradise is heaven. I will prove the latter to be the truth.
First we must note that Jesus Christ is God. God can be in more than one place at once.
In John 3:13, Jesus while on earth states that he is also in heaven.
 And no man hath ascended up to heaven, but he that came down from heaven, even the Son of man which is in heaven.
Secondly, the saints prior to the resurrection of Christ did indeed go to heaven.
2 King 2:1,11
-  And it came to pass, when the LORD would take up Elijah into heaven by a whirlwind, that Elijah went with Elisha from Gilgal.
-  And it came to pass, as they still went on, and talked, that, behold, there appeared a chariot of fire, and horses of fire, and parted them both asunder; and Elijah went up by a whirlwind into heaven.
 And Enoch walked with God: and he was not; for God took him.
 There was a certain rich man, which was clothed in purple and fine linen, and fared sumptuously every day:
 And there was a certain beggar named Lazarus, which was laid at his gate, full of sores,
 And desiring to be fed with the crumbs which fell from the rich man’s table: moreover the dogs came and licked his sores.
 And it came to pass, that the beggar died, and was carried by the angels into Abraham’s bosom: the rich man also died, and was buried;
 And in hell he lift up his eyes, being in torments, and seeth Abraham afar off, and Lazarus in his bosom.
 And he cried and said, Father Abraham, have mercy on me, and send Lazarus, that he may dip the tip of his finger in water, and cool my tongue; for I am tormented in this flame.
 But Abraham said, Son, remember that thou in thy lifetime receivedst thy good things, and likewise Lazarus evil things: but now he is comforted, and thou art tormented.
 And beside all this, between us and you there is a great gulf fixed: so that they which would pass from hence to you cannot; neither can they pass to us, that would come from thence.
 Then he said, I pray thee therefore, father, that thou wouldest send him to my father’s house:
 For I have five brethren; that he may testify unto them, lest they also come into this place of torment.
 Abraham saith unto him, They have Moses and the prophets; let them hear them.
 And he said, Nay, father Abraham: but if one went unto them from the dead, they will repent.
 And he said unto him, If they hear not Moses and the prophets, neither will they be persuaded, though one rose from the dead.
Next using the contextual dictionary of the scriptures, we can easily find out what paradise is. The word paradise is mentioned in the scriptures only three times.
The first instance is in the gospel of Luke.
 And one of the malefactors which were hanged railed on him, saying, If thou be Christ, save thyself and us.
 But the other answering rebuked him, saying, Dost not thou fear God, seeing thou art in the same condemnation?
 And we indeed justly; for we receive the due reward of our deeds: but this man hath done nothing amiss.
 And he said unto Jesus, Lord, remember me when thou comest into thy kingdom.
 And Jesus said unto him, Verily I say unto thee, To day shalt thou be with me in paradise.
Paul mentions paradise in 2 Corinthians 12.
2 Corinthians 12:1-5
 It is not expedient for me doubtless to glory. I will come to visions and revelations of the Lord.
 I knew a man in Christ above fourteen years ago, (whether in the body, I cannot tell; or whether out of the body, I cannot tell: God knoweth;) such an one caught up to the third heaven.
 And I knew such a man, (whether in the body, or out of the body, I cannot tell: God knoweth;)
 How that he was caught up into paradise, and heard unspeakable words, which it is not lawful for a man to utter.
Note the parallels: one caught up to the third heaven
he was caught up into paradise
Paul makes it very clear that the third heaven is paradise.
The third instance of the word paradise is found in Revelations 2.
 He that hath an ear, let him hear what the Spirit saith unto the churches; To him that overcometh will I give to eat of the tree of life, which is in the midst of the paradise of God.
Jesus states that the tree of life “is in the midst of the paradise”. So let’s find more information about the tree of life.
 And he shewed me a pure river of water of life, clear as crystal, proceeding out of the throne of God and of the Lamb.
 In the midst of the street of it, and on either side of the river, was there the tree of life, which bare twelve manner of fruits, and yielded her fruit every month: and the leaves of the tree were for the healing of the nations.
So the tree of life is in heaven. The tree of life is in paradise. So paradise must be heaven.