Hebrews 11:1

Now faith is the substance of things hoped for, the evidence of things not seen.
[Faith is the substance of things hoped for] Estin de pistis elpizomenoon hupostasis. Faith is the SUBSISTENCE of things hoped for; pragmatoon elengchos ou blepomenoon, The DEMONSTRATION of things not seen. The word hupostasis, which we translate "substance", signifies subsistence, that which becomes a foundation for another thing to stand on. And elengchos signifies such a conviction as is produced in the mind by the demonstration of a problem, after which demonstration no doubt can remain, because we see from it that the thing is; that it cannot but be; and that it cannot be otherwise than as it is, and is proved to be. Such is the faith by which the soul is justified; or rather, such are the effects of justifying faith: on it subsists the peace of God which passeth all understanding; and the love of God is shed abroad in the heart where it lives, by the Holy Spirit.
At the same time the Spirit of God witnesses with their spirits who have this faith that their sins are blotted out; and this is as fully manifest to their judgment and conscience as the axioms, "A whole is greater than any of its parts;" "Equal lines and angles, being placed on one another, do not exceed each other;" or as the deduction from prop. 47, book 1, Euclid: "The square of the base of a right-angled triangle is equal to the difference of the squares of the other two sides." Elengchos is defined by logicians, Demonstratio quae fit argumentis certis et rationibus indubitatis, qua rei certitudo efficitur. "A demonstration of the certainty of a thing by sure arguments and indubitable reasons." Aristotle uses it for a mathematical demonstration, and properly defines it thus: Elengchos de estin ho mee dunatos alloos echein, all' houtoos hoos heemeis legomen, "Elenchos, or Demonstration, is that which cannot be otherwise, but is so as we assert." Rhetor. ad Alexand., cap. 14, peri elengchou. On this account I have adduced the above theorem from Euclid.
[Things hoped for] Are the peace and approbation of God, and those blessings by which the soul is prepared for the kingdom of heaven. A penitent hopes for the pardon of his sins and the favour of his God; faith in Christ puts him in possession of this pardon, and thus the thing that was hoped for is enjoyed by faith. When this is received, a man has the fullest conviction of the truth and reality of all these blessings though unseen by the eye, they are felt by the heart; and the man has no more doubt of God's approbation and his own free pardon, than he has of his being.
In an extended sense the things hoped for are the resurrection of the body, the new heavens and the new earth, the introduction of believers into the heavenly country, and the possession of eternal glory. The things unseen, as distinguished from the things hoped for, are, in an extended sense, the creation of the world from nothing, the destruction of the world by the deluge, the miraculous conception of Christ, his resurrection from the dead, his ascension to glory, his mediation at the right hand of God, his government of the universe, etc., etc., all which we as firmly believe on the testimony of God's word as if we had seen them. See Macknight. But the faith has particular respect to the being, goodness, providence, grace, and mercy of God, as the subsequent verses sufficiently show.

From Adam Clarke's Commentary

Hebrews 11:1

Now faith is the substance of things hoped for, the evidence of things not seen.
Faith - in its widest sense: not restricted to faith in the Gospel. Not a definition of faith in its whole nature, but a description of its characteristics in relation to Paul's exhortation to perseverance (Heb 10:39).
Substance ... - it substantiates God's promises, the fulfillment of which we hope for, making them present realities to us. [However, hupostasis is translated in Heb 3:14, 'confidence.'] So Alford. Thomas Magister, like our version, 'The whole thing is virtually contained in the first principle; now the first commencement of the things hoped for is in us through the assent of faith, which virtually contains all the things hoped for' (cf. note, Heb 6:5). Through faith, the future object of Christian hope, in in its beginning, is already substantiated (Heb 11:6). Hugo de Victor distinguished faith from hope. By faith we are sure of eternal things that they ARE; by hope we are confident that WE SHALL HAVE them. Hope presupposes faith (Rom 8:25).
Evidence, [elengchos] - 'demonstration:' convincing proof to the believer; the soul thereby seeing what the eye cannot see.
Things not seen - the whole invisible spiritual world. 'Eternal life is promised to us, but it is when we are dead; we are told of a blessed resurrection, but meanwhile we moulder in the dust; we are declared to be justified, and sin dwells in us; we hear that we are blessed, meantime we are overwhelmed in miseries; we are promised abundance of all goods, but we still endure hunger and thirst; God declares He will immediately come to our help, but He seems deaf to our cries. What should we do if we had not faith and hope to lean on, and if our mind did not emerge amidst the darkness above the world by the shining of the Word and Spirit of God?' (Calvin.) Faith is an assent unto truths credible upon the testimony of God (not on their intrinsic reasonableness), delivered unto us in the writings of the apostles and prophets. Christ's ascension is the cause; His absence the crown of our faith (Dr. Pearson). Faith believes what it sees not; for if thou seest, there is no faith, the Lord has gone away so as not to be seen: He is hidden, that He may be believed; the yearning desire by faith after Him who is unseen is the preparation of a heavenly mansion for us; when He shall be seen, it shall be given to us as the reward of faith (Augustine). As revelation deals with invisible things exclusively, faith is the faculty needed by us. By faith we venture our eternal interests on the bare Word of God: this is altogether reasonable.

(from Jamieson, Fausset, and Brown Commentary