EXERCISES 6B

6C Normed Vector Spaces

Norms and Complete Norms

This section begins with a crucial definition.

6.33 Definition norm; normed vector space

Anormon a vector spaceV(overF) is a function∥·∥^:V→[0,∞)such that

• ∥f∥=0if and only iff =0(positive definite);

• ∥^αf∥=|^α| ∥f∥^{for allα}∈^Fandf ∈V(homogeneity);

• ∥f+_g∥ ≤ ∥f∥+∥g∥^{for all} f,g∈V(triangle inequality).

Anormed vector spaceis a pair(_V,∥·∥), whereVis a vector space and∥·∥^{is a} norm onV.

6.34 Example norms

• ^Supposen∈Z⁺. Define∥·∥1and∥·∥∞onFⁿby

∥(_a1, . . . ,an)∥1=|a1|+· · ·+|an| and

∥(_a1, . . . ,an)∥∞=max{|a1|^{, . . . ,}|an|}^. Then∥·∥1and∥·∥∞are norms onFⁿ, as you should verify.

• ^Onℓ¹(see the last bullet point in Example6.32for the definition ofℓ¹), define

∥·∥1by

∥(_a1,a2, . . .)∥1=

∑

∞ k=1|ak|^. Then∥·∥1is a norm onℓ¹, as you should verify.

• ^SupposeXis a nonempty set andb(_X)is the subspace ofF^Xconsisting of the bounded functions fromXtoF. For f a bounded function fromXtoF, define

∥f∥^by

∥f∥=sup{|f(_x)|^:x ∈X}^. Then∥·∥is a norm onb(X), as you should verify.

• LetC([0, 1])denote the vector space of continuous functions from the interval [0, 1]toF. Define∥·∥onC([0, 1])by

∥f∥= Z ₁

0 |f|^. Then∥·∥is a norm onC([0, 1]), as you should verify.

Sometimes examples that do not satisfy a definition help you gain understanding.

6.35 Example not norms

• LetL¹(R)denote the vector space of Borel (or Lebesgue) measurable functions f:R→Fsuch thatR

|f|^dλ<∞, whereλis Lebesgue measure onR[we are now modifying the definition ofL¹(_R)in3.45to allow for the possibility that F=C]. Define∥·∥1onL¹(_R)by

∥f∥1= Z

|f|^dλ.

Then∥·∥1satisfies the homogeneity condition and the triangle inequality on L¹(_R), as you should verify. However,∥·∥1is not a norm onL¹(_R)because the positive definite condition is not satisfied. Specifically, ifEis a nonempty Borel subset ofRwith Lebesgue measure0(for example,Emight consist of a single element ofR), then∥^χ_E∥1=0butχ_E̸=0. In the next chapter, we will discuss a modification ofL¹(R)that removes this problem.

• ^Ifn∈^Z+and∥·∥is defined onFⁿby

∥(_a1, . . . ,an)∥=|a1|^1/2+· · ·+|an|^1/2,

then∥·∥satisfies the positive definite condition and the triangle inequality (as you should verify). However,∥·∥as defined above is not a norm because it does not satisfy the homogeneity condition.

• ^If∥·∥1/2is defined onFⁿby

∥(_a1, . . . ,an)∥1/2= |a1|^1/2+· · ·+|an|^1/22,

then∥·∥1/2satisfies the positive definite condition and the homogeneity condi- tion. However, ifn>1then∥·∥1/2is not a norm onFⁿbecause the triangle inequality is not satisfied (as you should verify).

The next result shows that every normed vector space is also a metric space in a natural fashion.

6.36 normed vector spaces are metric spaces

Suppose(V,∥·∥)is a normed vector space. Defined:V×V→[0,∞)by d(_f,g) =∥f −g∥^.

Thendis a metric onV.

Proof Supposef,g,h∈V. Then

d(_f,h) =∥f−h∥=∥(_f −g) + (_g−h)∥

≤ ∥f−g∥+∥g−h∥

=_d(_f,g) +_d(_g,h).

Thus the triangle inequality requirement for a metric is satisfied. The verification of the other required properties for a metric are left to the reader.

From now on, all metric space notions in the context of a normed vector space should be interpreted with respect to the metric introduced in the previous result.

However, usually there is no need to introduce the metricdexplicitly—just use the norm of the difference of two elements. For example, suppose(_V,∥·∥)is a normed vector space, f1,f2, . . .is a sequence inV, and f ∈ V. Then in the context of a normed vector space, the definition of limit (6.8) becomes the following statement:

k→∞lim f_k = f means lim

k→∞∥f_k− f∥=0.

As another example, in the context of a normed vector space, the definition of a Cauchy sequence (6.12) becomes the following statement:

A sequence f1,f2, . . .in a normed vector space(V,∥·∥)is a Cauchy se- quence if for everyε>0, there existsn∈Z⁺such that∥f_j−f_k∥<εfor all integersj≥nandk≥n.

Every sequence in a normed vector space that has a limit is a Cauchy sequence (see6.13). Normed vector spaces that satisfy the converse have a special name.

6.37 Definition Banach space

A complete normed vector space is called aBanach space.

In a slight abuse of terminology, we often refer to a normed vector space Vwithout mentioning the norm∥·∥^. When that happens, you should

assume that a norm∥·∥lurks nearby, even if it is not explicitly displayed.

In other words, a normed vector space Vis a Banach space if every Cauchy se- quence inVconverges to some element ofV.

The verifications of the assertions in Examples6.38and6.39below are left to the reader as exercises.

6.38 Example Banach spaces

• The vector spaceC([0, 1])with the norm defined by∥f∥=sup

[0, 1]|f|is a Banach space.

• The vector spaceℓ¹with the norm defined by∥(_a1,a2, . . .)∥1=_∑^∞_k=1|a_k|^{is a} Banach space.

6.39 Example not a Banach space

• The vector spaceC([0, 1])with the norm defined by ∥f∥ = R₁

0|f| ^{is not a} Banach space.

• The vector spaceℓ¹with the norm defined by∥(_a1,a2, . . .)∥∞= sup

k∈Z⁺

|ak|^is not a Banach space.

6.40 Definition infinite sum in a normed vector space

Supposeg1,g2, . . .is a sequence in a normed vector spaceV. Then∑^∞_k=1g_kis defined by

∑

∞ k=1

gk = lim

n→∞

∑

n k=1

gk

if this limit exists, in which case the infinite series is said toconverge.

Recall from your calculus course that ifa1,a2, . . .is a sequence of real numbers such that∑^∞_k=1|ak| < _{∞, then∑}^∞_k=1akconverges. The next result states that the analogous property for normed vector spaces characterizes Banach spaces.

6.41

∑^∞_k=1∥g_k∥<∞ =⇒ ∑^∞_k=1g_kconverges

⇐⇒ Banach space SupposeVis a normed vector space. ThenVis a Banach space if and only if

∑^∞_k=1g_kconverges for every sequenceg1,g2, . . .inVsuch that∑^∞_k=1∥g_k∥<_∞.

Proof First supposeVis a Banach space. Supposeg1,g2, . . .is a sequence inVsuch that∑^∞_k=1∥gk∥ <_{∞. Suppose}ε> 0. Letn ∈ ^Z+ be such that∑^∞m=n∥gm∥< ε.

Forj∈^{Z+, let} fjdenote the partial sum defined by fj =_g1+· · ·+_gj. Ifk>_j≥n, then

∥fk− fj∥=∥gj+1+· · ·+_gk∥

≤ ∥gj+1∥+· · ·+∥gk∥

≤

∑

^∞

m=n∥gm∥

<ε.

Thus f₁,f2, . . .is a Cauchy sequence inV. BecauseVis a Banach space, we conclude that f1,f2, . . .converges to some element ofV, which is precisely what it means for

∑^∞_k=1g_kto converge, completing one direction of the proof.

To prove the other direction, suppose∑^∞_k=1g_k converges for every sequence g1,g2, . . .inVsuch that∑^∞_k=1∥g_k∥<_{∞. Suppose f1},f2, . . .is a Cauchy sequence inV. We want to prove that f1,f2, . . .converges to some element ofV. It suffices to show that some subsequence off1,f2, . . .converges (by Exercise14in Section6A).

Dropping to a subsequence (but not relabeling) and settingf0=0, we can assume

that ∞

k=1

∑

∥fk−fk−1∥<∞.

Hence∑^∞_k=1(_fk− f_k−1)converges. The partial sum of this series afternterms is fn. Thuslim_n→∞ fnexists, completing the proof.

Bounded Linear Maps

When dealing with two or more vector spaces, as in the definition below, assume that the vector spaces are over the same field (eitherRorC, but denoted in this book asF to give us the flexibility to consider both cases).

The notationT f, in addition to the standard functional notationT(f), is often used when considering linear maps, which we now define.

6.42 Definition linear map

SupposeVandWare vector spaces. A functionT:V→Wis calledlinearif

• T(_f +_g) =_{T f}+_Tgfor all f,g∈V;

• T(αf) =_{αT f} for allα∈^Fand f ∈V. A linear function is often called alinear map.

The set of linear maps from a vector spaceVto a vector spaceWis itself a vector space, using the usual operations of addition and scalar multiplication of functions.

Most attention in analysis focuses on the subset of bounded linear functions, defined below, which we will see is itself a normed vector space.

In the next definition, we have two normed vector spaces,VandW, which may have different norms. However, we use the same notation∥·∥for both norms (and for the norm of a linear map fromVtoW) because the context makes the meaning clear. For example, in the definition below, f is inVand thus∥f∥refers to the norm inV. Similarly,T f ∈Wand thus∥T f∥refers to the norm inW.

6.43 Definition bounded linear map;∥T∥^;B(_V,W)

SupposeVandWare normed vector spaces andT:V→Wis a linear map.

• The norm ofT, denoted∥T∥, is defined by

∥T∥=sup{∥T f∥^: f ∈Vand∥f∥ ≤¹}^.

• Tis calledboundedif∥T∥<_∞.

• The set of bounded linear maps fromVtoWis denotedB(_V,W).

6.44 Example bounded linear map

LetC([0, 3])be the normed vector space of continuous functions from[0, 3]toF, with∥f∥=sup

[0, 3]|f|^{. Define}T:C([0, 3])→C([0, 3])by (_{T f})(_x) =_x²_f(_x).

ThenTis a bounded linear map and∥T∥=9, as you should verify.

6.45 Example linear map that is not bounded

LetVbe the normed vector space of sequences(a₁,a2, . . .)of elements ofFsuch thata_k=0for all but finitely manyk∈Z⁺, with∥(a1,a2, . . .)∥∞=max_k∈Z⁺|a_k|. DefineT:V→Vby

T(_a1,a2,a3, . . .) = (_{a1, 2a2, 3a3}, . . .). ThenTis a linear map that is not bounded, as you should verify.

The next result shows that ifVandWare normed vector spaces, thenB(_V,W)is a normed vector space with the norm defined above.

6.46 ∥·∥is a norm onB(V,W)

SupposeV andW are normed vector spaces. Then ∥S+T∥ ≤ ∥S∥+∥T∥ and∥αT∥=|^α| ∥T∥^{for all}S,T ∈ B(_V,W)and allα∈ F. Furthermore, the function∥·∥is a norm onB(_V,W).

Proof SupposeS,T∈ B(_V,W). Then

∥S+_T∥=sup{∥(_S+_T)_f∥^: f ∈Vand∥f∥ ≤¹}

≤^sup{∥S f∥+∥T f∥^: f ∈Vand∥f∥ ≤¹}

≤^sup{∥S f∥^: f ∈Vand∥f∥ ≤¹}

+sup{∥T f∥^: f ∈Vand∥f∥ ≤¹}

=∥S∥+∥T∥^.

The inequality above shows that∥·∥satisfies the triangle inequality onB(_V,W). The verification of the other properties required for a normed vector space is left to the reader.

Be sure that you are comfortable using all four equivalent formulas for∥T∥^shown in Exercise16. For example, you should often think of∥T∥as the smallest number such that∥T f∥ ≤ ∥T∥ ∥f∥for all f in the domain ofT.

Note that in the next result, the hypothesis requiresWto be a Banach space but there is no requirement forVto be a Banach space.

6.47 B(_V,W)is a Banach space ifWis a Banach space

SupposeVis a normed vector space andWis a Banach space. ThenB(_V,W)is a Banach space.

Proof SupposeT1,T2, . . .is a Cauchy sequence inB(_V,W). If f ∈V, then

∥T_jf−T_kf∥ ≤ ∥T_j−T_k∥ ∥f∥,

which implies thatT1f,T2f, . . . is a Cauchy sequence inW. BecauseWis a Banach space, this implies thatT1f,T2f, . . . has a limit inW, which we callT f.

We have now defined a functionT:V→W. The reader should verify thatTis a linear map. Clearly

∥T f∥ ≤^sup{∥Tkf∥^:k∈^Z+}

≤ ^sup{∥T_k∥^:k∈Z⁺}∥f∥

for eachf ∈V. The last supremum above is finite because every Cauchy sequence is bounded (see Exercise4). ThusT∈ B(_V,W).

We still need to show thatlim_k→∞∥Tk−T∥=0. To do this, supposeε>0. Let n∈ ^Z+be such that∥Tj−Tk∥<εfor allj≥ nandk≥n. Supposej≥nand suppose f ∈V. Then

∥(_Tj−T)_f∥= lim

k→∞∥Tjf−Tkf∥

≤^ε∥f∥^. Thus∥T_j−T∥ ≤ε, completing the proof.

The next result shows that the phrasebounded linear mapmeans the same as the phrasecontinuous linear map.

6.48 continuity is equivalent to boundedness for linear maps

A linear map from one normed vector space to another normed vector space is continuous if and only if it is bounded.

Proof SupposeVandWare normed vector spaces andT:V→Wis linear.

First supposeTis not bounded. Thus there exists a sequencef1,f2, . . .inVsuch that∥f_k∥ ≤^{1for each}k∈Z⁺and∥T f_k∥ →∞ask→∞. Hence

k→∞lim fk

∥T fk∥ =0 and T fk

∥T fk∥

= ^{T fk}

∥T fk∥ ̸→^0,

where the nonconvergence to0 holds because T f_k/∥T f_k∥^{has norm1for every} k∈Z⁺. The displayed line above implies thatTis not continuous, completing the proof in one direction.

To prove the other direction, now supposeT is bounded. Suppose f ∈ Vand f1,f2, . . .is a sequence inVsuch thatlim_k→∞fk= _f. Then

∥T fk−T f∥=∥T(_fk−f)∥

≤ ∥T∥ ∥f_k− f∥^.

Thuslim_k→∞T f_k=_{T f}. HenceTis continuous, completing the proof in the other direction.

Exercise18gives several additional equivalent conditions for a linear map to be continuous.