William Barclay writes:
This passage gives us the perfect summary of the relationship between the Spirit of God and the spirit of man. Without the Spirit of Christ a man cannot begin to be a Christian at all (verse 9). It is the Spirit who makes him a son of God (verse 14), and who assures him of the reality of that sonship (verse 16). For the Christian man the Spirit must be the law of his life, his director, the standard by which he judges all things, the person whose gifts he most of all desires (verses 4, 5, 9). The Spirit brings him certain great gifts. The Spirit brings him liberation from the law of sin and death (verse 2). The man into whose life the Spirit has come is the man set free. The Spirit brings him peace (verse 6). The Spirit brings him victory in the warfare of the soul. The Spirit brings him life. His mortal body becomes alive with the life of Christ (verse 11). The non-Christian man without Christ and the Holy Spirit may be said to exist; he cannot be said to live. The Spirit brings him power (verse 13) and enables him to put to death the deeds of the body. The peace which the Spirit brings is the peace of conquest.By the Holy Spirit (Jesus' presence in and with us), in place of sin and death we have recieved life and peace. By the Holy Spirit, in place of condemnation we have become adopted heirs with Christ.
It may be said that for Paul the spirit of a man is the indwelloing of the power of God in that man, or, to put it in another way, it is the risen Christ resident within him. The spirit of man is that part of a man which has kinship with God, and which therefore gives a man fellowship with God and power to win the victory in the warfare of the soul.
You know, no one really "walks with Jesus." Jesus is at the right hand of glory. He promised, however, that He would never leave us or forsake us; He promised He'd be with us to the end of the age. He is with us by the Holy Spirit.
To say you walk with Jesus is to say that you have life and peace; true sonship to the Father.
To say you walk with Jesus is to say that your life evidences love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and self-control. To say you walk with Jesus is to say that fleshly desires and passions have been crucified and replaced with Spiritual desires and passions. (Galatians 5:22-25)
Life, peace, kinship, fruit, spiritual desires and passions--those things describe the life that "walks with Jesus" for those things reflect a life filled with the Holy Spirit.