This can sound like something trivial, but I keep seeing both of these being used towards the Hilbert Space, to the point that I am confused and not sure anymore. Even Wikipedia says that it's
"a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product",
So, we don't need a norm, we just use an inner product to induce a metric, right? Or we do need the norm? Can we have an inner product that gives us a metric, without inducing a norm?
In some other place in the very same article, Hilbert Space is called a Normed Space, even though it's not clear from its definition.
My problem is, that Normed Vector Spaces with an Inner Product are, in fact, quite rare. In general, you often have Normed Spaces, where inner product doesn't induce the norm.
So, how is this with Hilbert Space? Do an Inner Product Space with a metric must to be a Normed Space? If yes, then I can understand why it's called a Normed Space, but if there can be a Inner Product Space, with a metric coming from inner product, that doesn't induce a norm somewhere in between (is it possible?), then why people call Hilbert Space a Normed Space?
I would be glad, if someone could explain this and bring insight onto my confusion :-)