Now faith is the assurance of things hoped for, the conviction of things not seen. Indeed, by faith our ancestors received approval. By faith we understand that the worlds were prepared by the word of God, so that what is seen was made from things that are not visible.
By faith Abel offered to God a more acceptable sacrifice than Cain’s. Through this he received approval as righteous, God himself giving approval to his gifts; he died, but through his faith he still speaks.
By faith Enoch was taken so that he did not experience death, and “he was not found, because God had taken him.” For it was attested before he was taken away that “he had pleased God.” And without faith it is impossible to please him, for whoever would approach God must believe that he exists and that he rewards those who seek him.
By faith Noah, warned by God about events as yet unseen, respected the warning and built an ark to save his household; by this he condemned the world and became an heir to the righteousness that is in accordance with faith.
By faith Abraham obeyed when he was called to set out for a place that he was to receive as an inheritance, and he set out, not knowing where he was going. By faith he stayed for a time in the land he had been promised, as in a foreign land, living in tents, as did Isaac and Jacob, who were heirs with him of the same promise. For he looked forward to the city that has foundations, whose architect and builder is God.
By faith, with Sarah’s involvement, he received power of procreation, even though he was too old, because he considered him faithful who had promised. Therefore from one person, and this one as good as dead, descendants were born, “as many as the stars of heaven and as the innumerable grains of sand by the seashore.”
Faith - pistis in Greek. I prefer to translate this as "allegiance" or "trust." It could also mean "faithfulness" - the act of being faithful. So the verse could read: "Allegiance/faithfulness is confidence in things hoped for or expected."
Or again: "To be allegiant is to act in confidence that what you expect will actually happen; to be convinced that what you haven't seen will indeed happen."
Verse 3, by the way, is an example of aionios being translated accurately in the New Testament. The NRSV says: "By faith we understand that the worlds were prepared by the word of God." Worlds is aionios. Translators recognize that translating it as "eternity" would make no sense here. So something like "worlds," or perhaps "ages" or "aeons" makes more sense. Now they just need to catch up and consistently do this throughout the rest of the New Testament!
The Hebrew Bible examples that follow are being used to convince the reader that pistis - allegiance, faith, trust - as opposed to sacrifice or legal compliance, is what leads to approval from God.
Notice that these first examples are all pre-Moses and pre-Torah. Abel, Enoch, Noah, Abraham - they all had God's approval not because of Sinaic law (which didn't exist yet), but because of their allegiance, faith, and trust in God.