People often say LOTR is a story about hope. (I'm reminded of it because someone said it in the notes of my Faramir post.) And that's true, but it's not the whole picture: LOTR is in large part a story about having to go on in the absence of hope.
Frodo has lost hope, as well as the ability to access any positive emotion, by Return. He is already losing it in Towers: he keeps going through duty and determination and of course Sam's constant help.
For most of the story, Sam is fueled by hope, which is why it's such a huge moment when he finally lets go of the hope of surviving and returning home, and focuses on making it to the Mountain. To speed their way and lighten the load, he throws his beloved pots and pans into a pit, accepting that he will never cook, or eat, again.
When Eowyn kills the Witch King, she's beyond hope and seeking for a glorious death in battle. It's possible that in addition to her love and loyalty for Théoden, she's strengthened by her hopelessness, the fear of the Nazgúl cannot touch someone who's already past despair.
Faramir is his father's son, he doesn't have any more hope of Gondor's victory or survival than Denethor does, he says as much to Frodo. What hope have we? It is long since we had any hope. ... We are a failing people, a springless autumn. He knows he's fighting a losing war and it's killing him. When he rejects the ring, he doesn't do it in the hope that his people can survive without it, he has good reason to believe they cannot. He acts correctly in the absence of hope.
Of course LOTR has a (mostly) happy ending, all the unlikely hopes come true, the characters who have lost hope gain what they didn't even hope for, and everyone is rewarded for their bravery and goodness, so on some level the message is that hope was justified. But the book never chastises characters who lost hope, it was completely reasonable of them to do so. Despair pushed Théoden and Denethor into inaction, pushed Saruman into collaboration, but the characters who despaired and held up under the weight of despair are Tolkien's real heroes.
(In an early draft of Return, Frodo and Sam receive honorary titles in Noldorin: Endurance beyond Hope and Hope Unquenchable, respectively. Then he cut it, probably because it was stating the themes of the entire book way too obviously, because this is what Tolkien cared about, really: enduring beyond hope. Without hope.)
Also, people who know more than me about the concept of estel, feel free to @ me.
I agree with everything in the above post, and think that the knife's edge balance between despair and hope in LOTR is a really important theme when you see the characters that fall into despair and the ones that don't, and the ones that despair but don't fall.
But the thing about estel is that it's not the only elvish word for hope. There's amdir, which is hope with a foundation, it's based on something. Estel is hope without that foundation, based on nothing at all--a fool's hope, if you will. It's what drives the entire Quest, because sending the Ring to Mt Doom is objectively insane and there is no reason to believe that it'll work beyond like, Elrond saying so (and if anyone is familiar with estel, it's Elrond). But it does work, because everyone in the Fellowship comes together to make sure that it will, because they're clinging to that estel with their last fingernails and they aren't letting go. Even if they give up hope for themselves or for their homes for for their loved ones.
Anyway all that to say I just wanted to highlight this passage from ROTK:
Far above the Ephel Dúath in the West the night-sky was still dim and pale. There, peeping among the cloud-wrack above a dark tor high up in the mountains, Sam saw a white star twinkle for a while. The beauty of it smote his heart, as he looked up out of the forsaken land, and hope returned to him. For like a shaft, clear and cold, the thought pierced him that in the end the Shadow was only a small and passing thing: there was light and high beauty for ever beyond its reach. His song in the Tower had been defiance rather than hope; for then he was thinking of himself. Now, for a moment, his own fate, and even his master's, ceased to trouble him. He crawled back into the brambles and laid himself by Frodo's side, and putting away all fear he cast himself into a deep untroubled sleep.
That's the hope that people are talking about when they say LOTR is about hope, I think. It's a hope that's bigger than any single character, and it's so hard and at times impossible to keep holding on to, but in the end it wins out, in spite of everything.
