Kenneth Kuttler

Chapter 13

NormsIn this chapter, X and Y are finite dimensional vector spaces which have a norm. Thefollowing is a definition.

Definition 13.0.1 A linear space X is a normed linear space if there is a norm defined onX, ||·|| satisfying

||x|| ≥ 0, ||x|| = 0 if and only if x = 0,

||x+ y|| ≤ ||x||+ ||y|| ,

||cx|| = |c| ||x||

whenever c is a scalar. A set, U ⊆ X, a normed linear space is open if for every p ∈ U,there exists δ > 0 such that

B (p, δ) ≡ {x : ||x− p|| < δ} ⊆ U.

Thus, a set is open if every point of the set is an interior point. Also, limn→∞ xn= x meanslimn→∞ ∥xn − x∥ = 0. This is written sometimes as xn → x.

Note first that∥x∥ = ∥x− y + y∥ ≤ ∥x− y∥+ ∥y∥

so∥x∥ − ∥y∥ ≤ ∥x− y∥ .

Similarly∥y∥ − ∥x∥ ≤ ∥x− y∥

and so|∥x∥ − ∥y∥| ≤ ∥x− y∥ . (13.1)

To begin with recall the Cauchy Schwarz inequality which is stated here for conveniencein terms of the inner product space, Cn.

Theorem 13.0.2 The following inequality holds for ai and bi ∈ C.∣∣∣∣∣n∑

i=1

aibi

∣∣∣∣∣ ≤(

n∑i=1

|ai|2)1/2( n∑

i=1

|bi|2)1/2

. (13.2)

Let X be a finite dimensional normed linear space with norm ||·|| where the field ofscalars is denoted by F and is understood to be either R or C. Let {v1,· · · ,vn} be a basisfor X. If x ∈ X, denote by xi the i

th component of x with respect to this basis. Thus

x =

n∑i=1

xivi.

Definition 13.0.3 For x ∈ X and {v1, · · · ,vn} a basis, define a new norm by

|x| ≡

(n∑

i=1

|xi|2)1/2

where

x =

n∑i=1

xivi.

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Chapter 13NormsIn this chapter, X and Y are finite dimensional vector spaces which have a norm. Thefollowing is a definition.Definition 13.0.1 A linear space X is a normed linear space if there is a norm defined onX, ||-|| satisfying\Ix|| > 0, |[x|| =0 #f and only if x =0,IIx + yl] < |b<ll + Ilyll,Ilex|] = lel |||whenever c is a scalar. A set, U C X, a normed linear space is open if for every p © U,there exists 6 > 0 such thatB(p,6) = {a : |la —pl| <6} CU.Thus, a set is open if every point of the set is an interior point. Also, limp Xnjn= X meanslimp soo ||Kn — X|| = 0. This is written sometimes as Xn — xX.Note first thatIlx|| = lx —y + yl] < |lk- yl] + lly|lsoIIx|| — |lyll < llk-yl].Similarlylll] — IIx < Ix —y|and soillxIl — Ilylll S lx — yl]. (13.1)To begin with recall the Cauchy Schwarz inequality which is stated here for conveniencein terms of the inner product space, C”.Theorem 13.0.2 The following inequality holds for a; and b; € C.n n 1/2 n 1/2i=1 i=l i=1Let X be a finite dimensional normed linear space with norm ||-|| where the field ofscalars is denoted by F and is understood to be either R or C. Let {vi,---,v,,} be a basisfor X. If x € X, denote by 2; the i*” component of x with respect to this basis. Thusnx = ) Livii=1Definition 13.0.3 For x € X and {v1,---,Vn} a basis, define a new norm byh 1/2|x| = (>: Hs?) ,i=1wherenmx = ) LiVi-i=l325