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While vector spaces have additional structure compared to a metric space, a general vector space has no notion of “length” or “distance.”
A vector space together with a norm is called a normed vector space (or normed linear space ).
Note that any normed vector space is a metric space with induced metric $d(x,y)=\u2225x,-,y\u2225$ . (This follows since $\u2225x,-,y\u2225=\u2225x,-,z,+,z,-,y\u2225\in \u2225x,-,z\u2225+\u2225y,-,z\u2225$ .) While a normed vector space “feels like” a metric space, it is important to remember that it actually satisfies a great deal of additional structure.
Technical Note: In a normed vector space we must have (from N2) that $x=y$ if $\u2225x,-,y\u2225=0$ . This can lead to a curious phenomenon when dealing with continuous-time functions. For example, in ${L}_{2}\left([a,b]\right)$ , we can consider a pair of functions like $x\left(t\right)$ and $y\left(t\right)$ illustrated below. These functions differ only at a single point, and thus ${\parallel x\left(t\right)-y\left(t\right)\parallel}_{2}=0$ (since a single point cannot contribute anything to the value of the integral.) Thus, in order for our norm to be consistent with the axioms of a norm, we must say that $x=y$ whenever $x\left(t\right)$ and $y\left(t\right)$ differ only on a set of measure zero. To reiterate $x=y\nLeftrightarrow x\left(t\right)=y\left(t\right)\forall t\in [a,b]$ , i.e., when we treat functions as vectors, we will not interpret $x=y$ as pointwise equality, but rather as equality almost everywhere .
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