'The righteous shall live by faith'
The Hebrew emunah read as trust or as steadfastness (the Qumran commentary reads it of Torah-keeping) and Paul's gospel proof text.
Habakkuk 2:3-4
3For the vision is yet for the appointed time, and it hurries toward the end, and won’t prove false. Though it takes time, wait for it; because it will surely come. It won’t delay.
4Behold, his soul is puffed up. It is not upright in him, but the righteous will live by his faith.
Romans 1:17
17For in it is revealed God’s righteousness from faith to faith. As it is written, “But the righteous shall live by faith.”
What each tradition says
Rabbinic / Midrashic
Rashi applies “but the righteous shall live by his faith” to King Jeconiah, whom the wicked man is destined to exile. His righteousness shall stand for him. On the day this one is cast out of his grave, Evil-merodach shall raise Jehoiachin’s head and place his throne above the thrones of the kings.
Rashi (Rabbi Solomon ben Isaac, 1040-1105), Rashi on Habakkuk 2:4, in The Judaica Press Complete Tanach with Rashi (trans. A. J. Rosenberg), via the Sefaria API — comment on Habakkuk 2:4 ('but the righteous shall live by his faith')
Patristic
Augustine states that the righteousness of God was veiled in the Old Testament and revealed in the New. It is called the righteousness of God because by His bestowal of it He makes us righteous, just as salvation is called the Lord's because He makes us safe.
Augustine of Hippo (354-430), On the Spirit and the Letter, in Nicene and Post-Nicene Fathers, ser. 1, vol. 5 (ed. Philip Schaff; New York: Christian Literature Publishing Co., 1886) — ch. 18 [XI] (on Romans 1:16-17 and Habakkuk 2:4)
Catholic
Gigot holds that Habakkuk 2:4 contains the book's central and distinctive teaching. Its oracular form points to a principle of wider import, a general law of God's providence in governing the world: the wicked carries in himself the germs of his own destruction, while the believer carries those of eternal life.
Francis E. Gigot (1859-1920), The Catholic Encyclopedia, s.v. 'Habacuc (Habakkuk)' (New York: Robert Appleton Company, 1910) — section 'Prophetical teaching'
Orthodox
Chrysostom reads Habakkuk 2:4 as Paul’s confirmation from the Prophets: “the just shall live by faith.” He says it is not in the nature of things for one who is to live to live otherwise than by faith, and that Habakkuk speaks of the life to come.
John Chrysostom (c. 349-407), Homilies on the Epistle to the Romans, in Nicene and Post-Nicene Fathers, ser. 1, vol. 11 (ed. Philip Schaff; T&T Clark, Edinburgh, 1889) — Homily II, on Romans 1:17
Reformed / Protestant
Calvin notes some read amunat as truth or integrity, as if the prophet meant the just man is safer in his faithfulness and pure conscience than worldly people are in all their defenses. But Calvin says this frigidly extenuates the Prophet’s declaration, because they do not understand the righteousness of faith from which salvation proceeds.
John Calvin (1509-1564), Commentaries on the Twelve Minor Prophets, vol. 4: Habakkuk, Zephaniah, Haggai (trans. John Owen; Edinburgh: Calvin Translation Society, 1848) — commentary on Habakkuk 2:4
Evangelical
Wesley identifies the one lifted up as proudly contending with divine providence and relying on his own wit for safety. The just person is humble and upright, adoring divine providence and persuaded of God's promises. To live means to support himself by firm expectation of Zion's deliverance.
John Wesley (1703-1791), Explanatory Notes upon the Old Testament (1765), in Wesley's Notes on the Bible (CCEL) — note on Habakkuk 2:4