Here, honestly, there are basically only two possible interpretations:
〜に望む = to expect/hope for (something) from ~
生涯に = in (someone’s) lifetime
The first interpretation looks very sensible, and is also what a lot of people answering on HiNative mentioned, but the problem is that 望み was used, so I don’t think – though I could be wrong – that 生涯に is limited to only the first verb, and probably is also relevant to 知り得る. That aside, while looking 生涯 up, I found examples that used it to mean ‘in (someone’s) lifetime’ even when the verb used wasn’t something like いる or ある. That’s when I decided it had to be something more general.
EDIT: Hm… honestly though, maybe both interpretations are possible. But then there’s the bit about travelling the world, which would be something that would take a lifetime, so 生涯 probably should apply to both…? I think travelling around a lot would definitely increase the number of things one would know about more than the things one could hope for, at the least.
This – in my opinion – actually just comes from English. It’s a very tempting interpretation, and I wanted to use it too, because ‘to hope to know’ (or just ‘to hope to do’ in general) is a very pleasant, poetic phrase in English. The problem is that the Japanese sentence doesn’t support that, because the ‘knowing’ bit (知り) doesn’t appear as part of an object or object phrase for 望む. That means that the knowledge we’re talking about isn’t being hoped for, and that means we can’t use ‘hoping to know’ as a translation.
The quick and short version is that this is the form that allows you to place other verbs or adjectives after the verb or adjective you’re using, and it’s often similar to the て-form in meaning. It’s used for making compound verbs, and can also function as a noun (with the major exception of old-fashioned phrases like 見るがいい, in which case it’s the 連体形 – not the 連用形 – that is working as a noun). There isn’t that much else you need to know, other than the fact that it also can be combined with various suffixes, which are often helper verbs (助動詞) like たい or なさい, but most of this is probably grammar you already know without the technical details.
Now, I know that there technically is a slight difference between 連用形 alone and the て-form, which is actually (historically) 連用形+て, but I haven’t found the time or motivation to finish reading the studies I’ve found on it. The one thing that I’ve noticed is that with the 連用形, there tends to be more of a break in the sentence, and the two actions feel more distinct, whereas with the て-form, I tend to feel like the actions are more closely linked and more likely to happen one after the other. For the 連用形, this is called the 中止法, which definitely seems to express that idea, and according to 精選版 日本国語大辞典, the actions described can be expressed either in the order of occurrence, or without any particular order in mind (which is the case here, I think).
If you want to read the study I’ve found, it’s here: https://hermes-ir.lib.hit-u.ac.jp/hermes/ir/re/8516/ryugaku0000900150.pdf
But it’s about 11 pages of Japanese, so I won’t blame you if you don’t feel like going through all of it. (I think the furthest I’ve gone in a study is about 7 pages, with lots of little bits skipped in between.) Still, the section on something like 連用形の二つの形 might be helpful. (The page is no longer loading for me, so I’m relying on my memory of what I saw just now. I hope I didn’t get the section title wrong.)