In reply to annamolly1504:
So as others have pointed out... you need to define the ski-touring.
Basically ice-climbing is nearly available everywhere in Norway (conditions depending, naturally).
Ski resorts are more or less known (especially if you like bigger resorts and good skiing or at least good skiing infra). Albeit some of the miniscule resorts with only a lift of two, might actually offer the best skiing experience (good amount of untracked terrain, and no one else around... if that is your thing).
So The big hubs for downhill skiing are Trysil (biggest resort in Norway), Hemsedal and the few resorts close to Lillehammer. But Narvik Ski resort is not bad either (smaller, but really good terrain). The smaller resorts... IDK, as said, they might have really good skiing... but services and amount of terrain might be limited... and those that now... well, ain't talking... but I have heard of rumors.
For ice climbing, Hemsedal has even it's own guidebook... From Lillehammer, you need to drive, but I'm fairly certain that within 1-2h drive you will have more ice that you can shake a stick at. Trysil might be a was on that regards. Narvik has enough ice for a lifetime, the super classic Henrikkefossen (WI-4) is around an hour away, and I'm sure there is stuff closer.
As for touring, if you mean more in the sense of xc-skiing (but slower), then stuff like Rondane a bit north of Lillehammer would prolly be suited for both Lillehammer and also Trysil. For Hemsedal, I'm not sure, but I'm fairly certain again a bit of a drive will get you to vast wilderness with most likely a hut or two (manned or not, IDK) for some touring. Fairly certain you have something like that in Narvik vicinity.
And if you mean more like alpine ski-touring/-mountaineering... well, that might be a tad harder to accomplish... as a lot of the huts are on the flat fjell-touring terrain... so either the long distances are gonna suck on them alpine-touring skis... or the downhill portion on nordic touring rigs .
Narvik might actually have the best "option" for this... around 1h east... so around Riksgränsen or Abisko (in SWE). I'd say that depending on your conditions/ambitions, 4 or 5 full days tour might be enough to get you to the top of Kebnekaise (highest peak in Sweden) and back to civilization. So drop car in Abisko, and with train+bus (via Kiruna, perhaps a taxi as well) get to Nikkaluokta. From there it's a day to Tarfalla-hut, or Kebnekaise Fjellstation. Following day to the summit. And depending on skills, conditions the next few days head northwards via the Huts on Kungsleden... to reach Abisko (where your car awaits). Bonus is, that if the weather is faul in around Narvik... changes are, it'll be good in Abisko or Riskgränsen. And both offer also ski resorts (Riksgränsen is really good). And the day touring potential around there is amazing... and there's even ice climbing.