Well, much of my previous post was about Hebrews 11:1, and my point was that the author was not answering a general question ("What is faith?"). There is a context to this statement.
The author addresses long-time Christian believers who have already endured hardship for their faith and calls them to perseverance (10:33ff):
But recall those earlier days when, after you had been enlightened, you endured a hard struggle with sufferings, sometimes being publicly exposed to abuse and persecution, and sometimes being partners with those so treated. For you had compassion for those who were in prison, and you cheerfully accepted the plundering of your possessions, knowing that you yourselves possessed something better and more lasting. Do not, therefore, abandon that confidence of yours; it brings a great reward. For you need endurance, so that when you have done the will of God, you may receive what was promised.
To back up the exhortation to perseverance in faith he quotes approximately Habaqquq 2:3f:
For yet "in a very little while,
the one who is coming will come and will not delay;
but my righteous one will live by faith.
My soul takes no pleasure in anyone who shrinks back."
And comments on a few keywords of the quotation:
But we are not among those who shrink back and so are lost, but among those who have faith and so are saved (literally, for soulpossession).
Then he introduces a new development on the extraordinary value of that faith they already have:
Now (de) faith is the assurance (or essence, hupostasis) of things hoped for, the conviction (elegkhos) of things not seen.
It is not an explanation (especially not to "unbelievers") of what faith in general is or what it would take to believe if you don't. It is a particular interpretation of what a very specific (Christian) faith means and is worth to those who already share it, within a particular interpretative frame or "world view" which opposes heavenly/spiritual/eternal invisible realities to the earthly/physical/transient visible world of shadows (after the pattern of Plato's cave allegory, but applied in a completely different way of course). From this perspective "faith" is a token of the higher plane within the lower one, a spark of true light in a world of shadow. Perhaps what comes closest to the gist of this notion in Paul is the financial metaphor of arrabôn, "deposit, guarantee, warranty, first installment" in 2 Corinthians 1:22. In sum: an interesting theological concept, but totally irrelevant to a debate between "believers" and "unbelievers" about the existence or non-existence of "God". Imo.