Faith is the assured expectation of what is hoped for, the evident demonstration (footnote: Or "convincing evidence") of realities that are not seen.
Here is the entry in BDAG:
ἔλεγχος, ου, ὁ (Pind., Pre-Socr. et al.; pap, LXX; PsSol tit. [no. 9]; TestJos 6:6; Philo [s. CBarrett, JTS n.s. 1, ’50, 9], Joseph., Just., Tat., Ath.) ① the act of presenting evidence for the truth of someth., proof, proving (Pla., Gorg. 471e; Demosth. 44, 15 τὸ πρᾶγμα τὸν ἔλεγχον δώσει; Epict. 3, 10, 11; POxy 237 VIII, 17; PStras 41, 6 ὁ ἔλ. δεικνύσει; Job 23:7, cp. vs. 4; Philo, Praem. 4 ἔ. τ. ἀληθείας; Jos., Bell. 1, 626; 628, C. Ap. 2, 17) ἔ. πραγμάτων οὐ βλεπομένων faith is a proving (or conviction about) unseen things=faith means to be sure about things unseen (in contrast to confidence in the temporal) Hb 11:1. ② the act of charging a pers. with wrongdoing, accusation σύ μου ἔ. εἶ; will you accuse me? Hv 1, 1, 6 (other interpretations: complaint [Hab 2:1]: will you lodge a complaint against me?; conviction [BGU 1138, 13=Mitt-Wilck. II/2, 100, 13 (19/18 B.C.)]; Kaibel 814 of a monument that serves as a touchstone to identify either those well-disposed toward the dead or the impious; cp. TestJos 6:6). ③ expression of strong disapproval, reproof, censure, correction (Job 6:26; 13:6; Wsd 2:14; Philo, Rer. Div. Her. 76) 1 Cl 57:4f (Pr 1:25, 30); 2 Ti 3:16 v.l.—DELG s.v. ἐλέγχω. M-M. TW
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 315). Chicago: University of Chicago Press.
Here is the Greek:
Westcott and Hort / [NA27 variants]
Ἔστιν δὲ πίστις ἐλπιζομένων ὑπόστασις, πραγμάτων ἔλεγχος οὐ βλεπομένων·
Specifically, is "evident demonstration" (or "convincing evidence") an acceptable translation of ἔλεγχος? Other translations have "conviction" (e.g., ESV, NASB).