Unto the church of God which is at Corinth, to them that are sanctified in Christ Jesus, called to be saints, with all that in every place call upon the name of Jesus Christ our Lord, both theirs and ours: The Church of God which is at Corinth - This Church was planted by the apostle himself about a.d. 52, as we learn from Acts 18:1 (note), etc.
Sanctified in Christ Jesus - Ἡγιασμενοις, Separated from the corruptions of their place and age.
Called to be saints - Κλητοις ἁγιοις, Constituted saints, or invited to become such; this was the design of the Gospel, for Jesus Christ came to save men from their sins.
With all that in every place, etc. - All who profess Christianity, both in Corinth, Ephesus, and other parts of Greece or Asia Minor; and by this we see that the apostle intended that this epistle should be a general property of the universal Church of Christ; though there are several matters in it that are suited to the state of the Corinthians only.
Both theirs and ours - That is, Jesus Christ is the common Lord and Savior of all. He is the exclusive property of no one Church, or people, or nation. Calling on or invoking the name of the Lord Jesus, was the proper distinguishing mark of a Christian. In those times of apostolic light and purity no man attempted to invoke God but in the name of Jesus Christ; this is what genuine Christians still mean when they ask any thing from God for Christ's Sake.