|Stupid naming conventions. This post should so be called|
"The Falcon, the Mockingbird, and the Cuckoo".
(TV spoilers follow; book ones don't.)
That Lysa is a captive is never really in doubt; her castle has been taken. Which I don't mean as a metaphor. History may claim the Eyrie is impregnable, but think about Petyr's spiel to Sansa as they pass through the Bloody Gates about the sheer impossibility of an army taking the Arryn's fortress: "One man can be worth ten thousand". Littlefinger's two biggest weaknesses have always been Cat/Sansa and his need to show off how clever he is. The combination of the two is irresistible, and so he lets (his self-aggrandising version of) the truth out: he has taken the Eyrie. He, alone and unarmed, has done what no-one could in a thousand years; he has taken control of the impenetrable castle atop the Giant's Lance. He rules the Vale now, and Lysa and Robin are merely hostages.
Not that they see themselves as such, of course. Theon doesn't either, of course, but whilst Ramsay had to rely on Stockholm Syndrome to gain the love of his prisoner, Littlefinger had it from the very beginning. It's not entirely clear at what point Lysa bought into Team Petyr entirely, but it certainly was prior to her murdering her own husband and blaming the Lannisters prior to the series beginning. And it's gone deep, too. When Petyr reveals his guiding principle as being "destroy to create", it's immediately clear that Lysa and Robin have internalised this idea completely. Robin is still young, so his version of it is appropriately small scale and literal - finding a reason for Sansa to slap him might have been justification in and of itself, but nevertheless there was more to it - but Lysa too has embraced the philosophy, hence why she murdered her husband and set about a chain of events that got her sister (amongst others) brutally murdered.
Ultimately, of course, it got her killed. But that's not the only reason to feel sorry for Lysa, despite her crimes. Hoster Tully's younger daughter is one more victim of George RR Martin's favourite trick; a fantasy cliche thrust helplessly into the "real world". Lysa Tully was the classic princess in distress, forcibly betrothed to a much older man, whilst the love of her life was banished after their one night of passion together . In a thousand other stories, Lysa would have been saved from her terrible fate of marrying a kind, honest rich man who happened to be old and unattractive . But this is Martin, so no prince arrived at the eleventh hour to save her from her unwanted marriage. Instead, Lysa found herself locked in the inescapable high tower fairy tales seem so beloved of, waiting for the one man who could reach her where entire armies had failed.
And again, because this is Martin, what she got was Littlefinger. One bird swapped for another, and happiness in truth no less far away.
It's worth noting at this point that Lysa is more or less unique amongst Westerosi women in arranged marriages in that she never at any point felt happy in her marriage. Cat and Dany both started off unhappy at the choices their family had pressed them into, but learned to love their husbands right up until said men were killed. Cersei ended up in a marriage that was almost always unhappy and ended up abusive, but she herself admits she felt something for Robert for quite some time after their wedding. Olenna Tyrell actually weaselled her way out of her intended betrothal, and Margaery - well, there's no reason to believe she actually loved Renly and plenty of reason to believe she hated Joffrey, but at least neither match lasted very long. Poor Lysa had to suffer almost two decades of marriage to a man she despised whilst the man she wanted to marry (who, unlike Cersei's lover, would have been entirely acceptable a mate if only he'd had more land and money) was denied her. In this way she ranks as the very best example in the whole series of a woman suffering as so many women must have suffered during our own history. No happy endings. No last-minute reprieves so that she could marry for love. No learning to understand that the man she was forced to marry and then sleep with was actually not so bad of a chap after all. Not even the properly miserable abusive marriage a story might present in order to invoke our sympathy more cheaply. Just a bog-standard, uneventful life of simmering misery.
And through it all, her only chance of escape is represented by Petyr Baelish, In a sense, Lysa's predicament might be closer to Theon's than we thought, in that for both of them their object of obsessive love is tightly bound up with the idea that those people represent: a sick form of authority and safety for Theon, the chance of escape from an utterly unsatisfying life for Lysa. Littlefinger is the gallant prince who will swoop in and take her away from it all (metaphorically speaking; she still wants the Eyrie because it keeps her son safe, and because the views are to die for). She's committed so totally to this fantasy that she's murdered her husband and betrayed her family to bring it into being. She cut herself off almost completely from her old family when she was forced into marriage (though TV watchers might be interested to hear Brynden "Blackfish" Tully was in charge of the Bloody Gates beneath the Eyrie until Cat kicked off the War of Five Kings), but she was never happy calling herself an Arryn.
Which doesn't mean the name didn't suit her. I wasn't entirely comfortable starting this essay from the perspective of Lysa representing House Arryn. It felt a bit too much like buying into the idea that the surname forced upon her defines her, despite the fact that unlike Cat she never identified herself with it. Ultimately though it became clear that I couldn't write up Lysa as a Tully, because she's put so much effort into first ignoring and then actively working against her own family for her to fit in to what would clearly have been the Tully storyline this season if there'd be time for it; the miserable horrors of having lost within two years a war, thousands of warriors and civilians, dozens of towns, three direct relations, and the major portion of your supplies for the coming winter. Lysa quite literally stands above and apart from all that. In contrast, she has completely embraced the actual facts of Arryn life, even if she hated her husband. The two qualities we've been repeatedly told Jon Arryn possessed were wisdom and honour. It's impossible to believe that when he was alive the Moon Door would be used except after deliberation and ceremony, much as how Eddard Stark executed the Night's Watch deserter so many episodes ago. Lysa on the other hand is clearly perfectly happy just dropping people through the hole whenever her mood suits her. It's just so fast and convenient and clean (well, it is for her; one hopes there's no-one living too close to the blast radius of condemned prisoners). It is, as Lysa's threats to Sansa demonstrated, something you can just swing open on the spur of the moment to press into service before you have time to reconsider. Which suits Lysa's mercurial, paranoid nature just fine, just as the Eyrie's inaccessibility satisfies her need for protection.
So as much as Lysa hated being called an Arryn, the assorted trappings of the role certainly seemed to suit her. And of course she was very useful as an Arryn to Littlefinger. Tragically, in the end, she was rather less useful as a Baelish. Decades of resentment and violent outbursts are really useful when you want someone to turn on the people they blame - fairly or not - for their problems. They are rather less useful when you've brought home another one of the resented ones, whom you rather want to survive. And it was clear Lysa resented Sansa. Even before she watched the kiss in the courtyard, when Lysa talked about Sansa's mother and Petyr's interest in Cat and Sansa both, you could almost hear the gears grinding away in her head. She resents Sansa because she's still young and pretty, because she escaped an unhappy forced marriage within months rather than decades, because she's a reminder of how happy Cat was with her life and how many children she bore, and lastly and most perversely, because she assumes Sansa is delighted at the prospect of wedding Robin, which is something else she has up on her aunt, who's only approaching a blissfully happy marriage comparatively late in life.
In the end, it was inevitable that something was going to snap. Nine times out of ten, the stupidest thing you can do when someone you care about seems to be drifting to someone else is to force them to choose. At the very best, you'll "win" at the cost of their resentment and regret, something you might have hoped Lysa might have been able to sympathise with. But in the end, the fantasy was more powerful than years of experience, and Lysa made her final mistake.
With his mother dead, Robin is the last Arryn, heir to the Vale just as Sansa is (so far as Littlefinger knows) heir to Winterfell. It's a nice little power base Littlefinger has assembled for himself. Sansa's potential is rather more theoretical than Robin's at present - whilst the Boltons rule the north there's a limit to how much mileage Petyr can get out of Ned Stark's last known surviving true-born child. But with Robin he has power now. All it took was one quick push - one slight woman failing to keep her balance - and after a thousand years no-one will any more hear the harsh cry of the Arryn falcon high above the Vale.
Well, they will. But that's the thing about mockingbirds, isn't it? They can sound like whatever bird they like...
 I said no book spoilers and I meant it, but some back-story for TV viewers might be useful here. The books aren't explicit on the subject, but from various comments from various characters it's been pieced together that following Petyr's duel with Brandon Stark, Lysa, already smitten, visited him to comfort him. Out of it either on pain meds or actual booze, Petyr mistook Lysa for Cat and slept with her. This led to Lysa falling pregnant and Petyr being kicked out of Riverrun. Lysa's father then had the child aborted through the use of tansy. The open secret that she had been besmirched in this way lowered her selling price as a bride - nice place, Westeros - but it turned out to actually be of benefit where Jon Arryn was involved, because Jon's first wife had provided him with no heir, so a woman of proven fertility was just what he needed.
Of course, ultimately Lysa only had one child with Jon. This has been taken by some to mean that the abortion of Lysa's pregnancy left her unable to bear many children. The far more likely explanation, I think, is that Jon himself was sterile, and Robin (Robert in the books) is actually Littlefinger's son. Right now, only Martin himself knows.
 I'm being facetious here. I don't mean to imply that Lysa's husband-to-be was cool and so she should shut the hell up. But at the same time it would be contrary to Martin's goals to not note that being forcibly married to a kind rich man is a vastly preferable fate to that handed to an awful lot of people in Westeros.