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Course by: Mark A. Davenport. E-mail the author

# Normed Vector Spaces

Module by: Mark A. Davenport. E-mail the author

While vector spaces have additional structure compared to a metric space, a general vector space has no notion of “length” or “distance.”

## Definition 1

Let VV be a vector space over KK. A norm is a function ·:VR·:VR such that

• N1. x 0 x V x 0 x V
• N2. x=0x=0 iff x=0x=0
• N3. αx=|α|xxVαx=|α|xxV, αKαK
• N4. x + y x + y x , y V x + y x + y x , y V

A vector space together with a norm is called a normed vector space (or normed linear space).

## Example 1

• V=RNV=RN: x2=i=1N|xi|2x2=i=1N|xi|2
• V=RNV=RN: x1=i=1N|xi|x1=i=1N|xi| (“Taxicab”/“Manhattan” norm)
• V=RNV=RN: x=maxi=1,...,N|xi|x=maxi=1,...,N|xi|
• V=Lp[a,b]V=Lp[a,b], p[1,)p[1,): x(t)p=ab|x(t)|pdt1/px(t)p=ab|x(t)|pdt1/p (The notation Lp[a,b]Lp[a,b] denotes the set of all functions defined on the interval [a,b][a,b] such that this norm exists, i.e., x(t)p<x(t)p<.)

Note that any normed vector space is a metric space with induced metric d(x,y)=x-yd(x,y)=x-y. (This follows since x-y=x-z+z-yx-z+y-zx-y=x-z+z-yx-z+y-z.) 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=yx=y if x-y=0x-y=0. This can lead to a curious phenomenon when dealing with continuous-time functions. For example, in L2([a,b])L2([a,b]), we can consider a pair of functions like x(t)x(t) and y(t)y(t) illustrated below. These functions differ only at a single point, and thus x(t)-y(t)2=0x(t)-y(t)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=yx=y whenever x(t)x(t) and y(t)y(t) differ only on a set of measure zero. To reiterate x=yx(t)=y(t)t[a,b]x=yx(t)=y(t)t[a,b], i.e., when we treat functions as vectors, we will not interpret x=yx=y as pointwise equality, but rather as equality almost everywhere.

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