Alford cites this phrase as one of the very few links which connect this epistle with the speech of James in Act 15:13.
The poor of this world (τοὺς πτωχοὺς τοῦ κόσμου)
But the correct reading is τῷ κόσμῳ, to the world; and the expression is to be explained in the same way as ἀστεῖος τῷ Θεῷ, fair unto God, Act 7:20, and δυνατὰ τῷ Θεῷ, mighty through (Rev., before) God, 2Co 10:4. So Rev., poor as to the world, in the world's esteem. Poor, see on Mat 5:3.
Rich in faith
The Rev., properly, inserts to be, since the words are not in apposition with poor, but express the object for which God has chosen them. Faith is not the quality in which they are to be rich, but the sphere or element; rich in their position as believers. “Not the measure of faith, in virtue of which one man is richer than another, is before the writer's mind, but the substance of the faith, by virtue of which every believer is rich” (Wiesinger, cited by Alford).