A.T. Robertson Commentary John 17:23

"I in them, and thou in me, that they may be perfected into one; that the world may know that thou didst send me, and lovedst them, even as thou lovedst me." — John 17:23 (ASV)

That they may be perfected into one (ινα ωσιν τετελειωμενο εις εν). Purpose clause again with ινα (nineteen times in this prayer, this the fifteenth) with the periphrastic perfect passive subjunctive of τελειοω (verse 4), permanent state, with εις εν (into one) as the goal and final result.

That the world may know (ινα γινωσκη). Present active subjunctive of γινωσκω with ινα like the present tense of πιστευω in verse 21, "that the world may keep on knowing" with the same pregnant phrase "that thou me didst send" (οτ συ με απεστειλας) as in 8,25.

And lovedst them (κα ηγαπησας αυτους). Timeless aorist, but love shown by sending Christ (John 3:16) and illustrated and proven by the way Christians love one another.