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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.”

Definition 1

Let V be a vector space over K . A norm is a function · : V R such that

  • x 0 x V
  • x = 0 iff x = 0
  • α x = | α | x x V , α K
  • x + y x + y x , y V

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

  • V = R N : x 2 = i = 1 N | x i | 2
    An illustration showing a point x in R2 and it's ell_2 (Euclidian) norm.  The norm is equal to the length of a straight line connecting x to the origin.
  • V = R N : x 1 = i = 1 N | x i | (“Taxicab”/“Manhattan” norm)
  • V = R N : x = max i = 1 , . . . , N | x i |
    An illustration showing a point x in R2 and it's ell_infinity norm.  The norm is equal to the length of the longer of the two (orthogonal) paths that connect x to the x- and y-axes.
  • V = L p [ a , b ] , p [ 1 , ) : x ( t ) p = a b | x ( t ) | p d t 1 / p (The notation L p [ a , b ] denotes the set of all functions defined on the interval [ a , b ] such that this norm exists, i.e., x ( t ) p < .)

Note that any normed vector space is a metric space with induced metric d ( x , y ) = x - y . (This follows since x - y = x - z + z - y x - 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 = y if x - y = 0 . This can lead to a curious phenomenon when dealing with continuous-time functions. For example, in L 2 ( [ a , b ] ) , we can consider a pair of functions like x ( t ) and y ( t ) illustrated below. These functions differ only at a single point, and thus x ( 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 = y whenever x ( t ) and y ( t ) differ only on a set of measure zero. To reiterate x = y x ( t ) = y ( t ) t [ 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 .

A smooth function defined on the interval [-1,1]. A function that is identical to the previous function, except for a point discontinuity where it takes a different value.

Questions & Answers

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Need help solving this problem (2/7)^-2
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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