Paul speaks of God showing that we belong to him as if God had put a mark on us as a sign that we belong to him. Alternate translation: "he has put his mark of ownership on us" or "he has shown that we belong to him" (See: [[rc://en/ta/man/translate/figs-metaphor]])
Here the word "hearts" refers to the innermost part of a person. Alternate translation: "gave us the Spirit to live within each of us" (See: [[rc://en/ta/man/translate/figs-metonymy]])