Christ was "a Son over His house whose house we are if we hold fast our confidence and the boast of our hope" (Heb 3:6). The word "hope" is used several times in the book of Hebrews. Having hope means that you have an expectation that something will come to pass. It can be based on justification or reasonableness, otherwise it would be hopeless. You can even have hope in a person who you feel can accomplish what you expect. Hope can have a connotation of doubt because if what you expect seems impossible, you wouldn't say "I hope so" questioning it. Therefore "hope that is seen is not hope; for who hopes for what he already sees?" (Ro 8:24). "But if we hope for what we do not see, with perseverance we wait eagerly for it" (:25). There is an eager anticipation not normally there as we are "looking for the blessed hope and appearing of the glory of our great God and Savior, Jesus Christ" (Titus 2:13). To Paul it was "according to my earnest expectation and hope" (Php 1:20).
One piece of the armor of God is the "helmet, the hope of salvation" (1Th 5:8). It protects the mind. When you get an understanding of something you reply, "I see what you mean." Seeing is a physical experience and a metaphor for mentally comprehending something. When it is established "our hope for you is firmly grounded, knowing" (2Co 1:7). It is a deliberate decision. Paul advised "instruct those who are rich in this present world not . . . to fix their hope on the uncertainty of riches" (1Ti 6:17). It shows you can misplace your hope as the Jews were told it is "'Moses, in whom you have set your hope'" (Jn 5:45). The proper focus is that "we have fixed our hope on the living God, who is the Savior of all men" (1Ti 4:10) and it is "He on whom we have set our hope. And He will yet deliver us" (2Co 1:10).
There are attributes which include "believes all things, hopes all things" (1Co 13:7). The two are separated. Also "now faith, hope, love, abide these three" (:13). Separate words have individual meanings which sometimes need careful interpretation. Abraham is said that "in hope against hope he believed" (Ro 4:18). Both hope and believing are cited incorporating two meanings. Hope is fixing or setting your mind on a probable future result. "Against hope" (:18) means you wouldn't logically expect something to happen because the chance of it is almost zero. Abraham nonetheless hoped in it anyway and "in hope . . . he believed" (:18). These two words must have different uses or else they'd be redundant together. For instance, "we through the Spirit, by faith, are waiting for the hope of righteousness" (Gal 5:5). "We have obtained our introduction by faith into this grace in which we stand; and exalt in hope of the glory of God" (Ro 5:2). Paul prayed that "the God of hope fill you with all joy and peace in believing, so that you will abound in hope by the power of the Holy Spirit" (15:13). We "through Him are believers in God, who raised Him from the dead and gave Him glory, so that your faith and hope are in God" (1Pe 1:21). Abraham "believed in hope" (Ro 4:18 KJV) because he trusted in the promise "that he might become a father of many nations according to that which had been spoken" (:18).