{Now faith is} (estin de pistis). He has just said that
"we are of faith" ( 10:39 ), not of apostasy. Now he proceeds in
a chapter of great eloquence and passion to illustrate his point
by a recital of the heroes of faith whose example should spur
them to like loyalty now. {The assurance of things hoped for}
(elpizomen(936e) hupostasis). {Hupostasis} is a very common word
from Aristotle on and comes from huphist(886d)i (hupo, under,
hist(886d)i, intransitive), what stands under anything (a building,
a contract, a promise). See the philosophical use of it in  1:3 ,
the sense of assurance (une assurance certaine, M(826e)(8267)oz) in
 3:14 , that steadiness of mind which holds one firm ( 2Co 9:4 ).
It is common in the papyri in business documents as the basis or
guarantee of transactions. "And as this is the essential meaning
in  Heb 11:1  we venture to suggest the translation 'Faith is the
_title-deed_ of things hoped for'" (Moulton and Milligan,
_Vocabulary_, etc.). {The proving of things not seen} (pragmat(936e)
elegchos ou blepomen(936e)). The only N.T. example of elegchos
(except Textus Receptus in  2Ti 3:16  for elegmon). Old and
common word from elegch(935c) ( Mt 18:15 ) for "proof" and then for
"conviction." Both uses occur in the papyri and either makes
sense here, perhaps "conviction" suiting better though not in the
older Greek.