Our everyday usage of "hope" includes a blend of temporal and spiritual. Some may say that they hope it snows tomorrow (or that it doesn't snow). Others might say that they hope to come into a large sum of money. In some situations, we might allege to hope that someone falls and breaks their neck. At other times, we hope that something we said did not hurt or offend another. All of these usages of "hope" can help us to find meaning in the way that the word hope appears in scripture.
From these everyday usages, we can find hope in things both future and past. Hope can apply to things that are morally good or bad. In addition, hope might lie in something with a remote possibility or in something with a significant likelyhood of obtaining. As we analyze this, we may conclude that hope does not necessarily lend itself well to a categorizational approach of definition. Although many times hope involves confidence in an outcome, it does not always involve such confidence (i.e., the hope to come into a large sum of money). Nevertheless, it seems that one could not truly hope for the impossible (i.e., saying "I hope that the lawn mows itself tomorrow" seems to mean more that "it would be cool if..." than a true hope that I'll arise in the morning to find the grass has mowed itself).
In this context, a necessary component of hope is belief that the outcome desired or hoped for is within the realm of possibility. From the example of hoping that something already said and heard did not offend another, we can see that hope does not have to be always forward-looking. Some hopes look forward, others look backward. We can truly hope that an accident victim who passed away did not suffer too much before passing. Similarly, we can hope that we will avoid a similar fate in the future.
As such, hope suggests an openness to possibilities, or to a desired possibility, that is probable to some degree or another. Under such an approach, Elder Maxwell's distinction between proximate hope and eternal hope can guide the connection between our understanding of hope and faith. Proximate hope centers on things connected to our current, mortal, horizon. We hope for safety for our family or for professional success. Eternal hope seems to add to the general understanding of hope the component of faith. When Paul refers to a "lively hope" it seems he may be alluding to the same idea. Hope takes on added life when coupled with a choice to accept or believe as true a worldview that assists us to orient ourselves toward God and others and that allows us to organize our lives accordingly.
Thus, faith seems to occupy the position of a belief-choice regarding a worldview. Faith does not imply knowledge (indeed, the two notions are considered distinct in the canon of scripture), but does imply a course of life predicated upon the choice to believe. At its root, any living in the world requires that we make certain belief-choices. We choose to believe in the connection between certain choices and consequences and reorder our lives based on this belief-choice. Similarly, we choose to accept a worldview including (or excluding) God's existence and reorient ourselves to our world, to ourselves, and to others as a result.
In this manner, faith becomes a stabilizing force in our lives, supplying us with assurance and evidence for the life we have built upon it. Hope occupies the role of an openness (tied to optimism) to the possibilities in our lives and worldviews. In this way, faith and hope intertwine, with belief-choices contributing to our worldviews and the limits of our possibilities and hope reinforcing our faith, motivating us to take the needed steps to reorient ourselves to the new world (kingdom) created through our belief choice. (And, interestingly, this reorientation, when premised upon the doctrines of Jesus Christ, of necessity involves a charitable reorientation to the world.) In sum, faith and hope do not express the same thing, but mutually reinforce one another (again, not excluding charity, also), in a manner that makes hope enlivening.
The perfect brightness of hope spoken of in holy writ comes through faith in the Lord Jesus Christ as our Messiah and Redeemer. This principle opens up eternal possibilities and creates for us a new identity and thrusts us into a new creation. To instill faith and hope, however, requires that our initial interest or desire be met with the Lord's Spirit to infuse that interest or desire with the requisite change in worldview and orientation to possibility that the result is that we become new creatures in Christ--new, faithful and hopeful persons. Thus, again, the beginning and end of our study turns and returns us to seeking and obtaining the Spirit of the Lord, which will fill us with faith, hope and charity.