Regarding Hebrews 11.1, I found this passage in Paul Ellingworth’s commentary on Hebrews helpful:

It is more natural, in the light of the chapter [Heb. 11] as a whole, to think of v. 1 as a summary of what faith does: faith binds the believer securely to the reality of what he does not (yet) see, but for which he hopes.  The difficulty of understanding the statement as definition is accordingly attenuated by translating ὑπόστασις and ἔλεγχος, and sometimes πίστις also, by verbal expressions: for example, “to have faith is to be sure of the things we hope for, to be certain of the things we cannot see” (TEV); “only [?] faith can guarantee the blessings that we hope for, or prove the existence of realities that are unseen” ([N]JB).  This has the effect of presenting the verse as a statement about the function rather than the nature of faith; this accords well with the rest of the chapter, which is mainly about action on the basis of faith (bold emphasis added).

Paul Ellingworth, The Epistle to the Hebrews, New International Greek Testament Commentary (Grand Rapids: William B. Eerdmans Publishing Company, 1993), 566.

That was a bit heavy in the middle, so let me summarize:

The context of the entire 11th chapter of Hebrews indicates that we should read verse 1 as a description of what faith does, not as a definition of what faith is.  Bible versions that translate the key Greek nouns (assurance, conviction, faith) in active terms reduce the difficulty of attempting to explain a very awkward “definition.”  Since the rest of the chapter tells what people did by faith, it makes sense to read the opening verse as a summary of what faith does.