Preface

参考摘录于FUNCTIONAL ANALYSIS NOTES——Mr. Andrew Pinchuck

1 Linear Spaces

1.1 linear space

A linear space over a field F\mathbb{F}F is a nonempty set XXX with two operations,F\mathbb{F}F denote either R\mathbb{R}R or C\mathbb{C}C
+:X×X→X(+: X \times X \rightarrow X \quad(+:X×X→X( called addition ),),), and

⋅\Large \cdot⋅   :F×X→X: \mathbb{F} \times X \rightarrow X \quad:F×X→X (called multiplication)
satisfying the following properties:

[1]x+y∈X[1] x+y \in X[1]x+y∈X whenever x,y∈Xx, y \in Xx,y∈X
[2] x+y=y+xx+y=y+xx+y=y+x for all x,y∈Xx, y \in Xx,y∈X
[3] There exists a unique element in X,X,X, denoted by 0 , such that x+0=0+x=xx+0=0+x=xx+0=0+x=x for all x∈Xx \in Xx∈X;
[4] Associated with each x∈Xx \in Xx∈X is a unique element in X,X,X, denoted by −x,-x,−x, such that x+(−x)=x+(-x)=x+(−x)= −x+x=0-x+x=0−x+x=0
[5] (x+y)+z=x+(y+z)(x+y)+z=x+(y+z)(x+y)+z=x+(y+z) for all x,y,z∈Xx, y, z \in Xx,y,z∈X

[6] α⋅x∈X\alpha \cdot x \in Xα⋅x∈X for all x∈Xx \in Xx∈X and for all α∈F\alpha \in \mathbb{F}α∈F
[7]α⋅(x+y)=α⋅x+α⋅y[7] \alpha \cdot(x+y)=\alpha \cdot x+\alpha \cdot y[7]α⋅(x+y)=α⋅x+α⋅y for all x,y∈Xx, y \in Xx,y∈X and all α∈F\alpha \in \mathbb{F}α∈F
[8](α+β)⋅x=α⋅x+β⋅x[8](\alpha+\beta) \cdot x=\alpha \cdot x+\beta \cdot x[8](α+β)⋅x=α⋅x+β⋅x for all x∈Xx \in Xx∈X and all α,β∈F\alpha, \beta \in \mathbb{F}α,β∈F
[9](αβ)⋅x=α⋅(β⋅x)[9](\alpha \beta) \cdot x=\alpha \cdot(\beta \cdot x)[9](αβ)⋅x=α⋅(β⋅x) for all x∈Xx \in Xx∈X and all α,β∈F\alpha, \beta \in \mathbb{F}α,β∈F
[10]1⋅x=x[10] 1 \cdot x=x[10]1⋅x=x for all x∈Xx \in Xx∈X
representation:(X,F,+,⋅)(X, \mathbb{F},+, \cdot)(X,F,+,⋅)

then a linear space is also called a vector space and its elements are called vectors

Examples

[1] Let X=C[a,b]={x:[a,b]→F∣xX=\mathcal{C}[a, b]=\{x:[a, b] \rightarrow \mathbb{F} \mid xX=C[a,b]={x:[a,b]→F∣x is continuous }. Define the operations of addition and scalar multiplication pointwise: For all x,y∈Xx, y \in Xx,y∈X and all α∈R,\alpha \in \mathbb{R},α∈R, define
(x+y)(t)=x(t)+y(t)and (α⋅x)(t)=αx(t)}for all t∈[a,b]\left.\begin{array}{ll} (x+y)(t) & =x(t)+y(t) \text { and } \\ (\alpha \cdot x)(t) & =\alpha x(t) \end{array}\right\} \text { for all } t \in[a, b] (x+y)(t)(α⋅x)(t)​=x(t)+y(t) and =αx(t)​} for all t∈[a,b]
Then C[a,b]\mathcal{C}[a, b]C[a,b] is a real vector space.

Sequence: Informally, a sequence in XXX is a list of numbers indexed by N.\mathbb{N} .N. Equivalently, a sequence in XXX is a function x:N→Xx: \mathbb{N} \rightarrow Xx:N→X given by n↦x(n)=xn.n \mapsto x(n)=x_{n} .n↦x(n)=xn​. We shall denote a sequence x1,x2,…x_{1}, x_{2}, \ldotsx1​,x2​,… by
x=(x1,x2,…)=(xn)1∞x=\left(x_{1}, x_{2}, \ldots\right)=\left(x_{n}\right)_{1}^{\infty} x=(x1​,x2​,…)=(xn​)1∞​
[2] The sequence space s. Let s denote the set of all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ of real or complex numbers. Define the operations of addition and scalar multiplication pointwise: For all x=x=x= (x1,x2,…),y=(y1,y2,…)∈s\left(x_{1}, x_{2}, \ldots\right), y=\left(y_{1}, y_{2}, \ldots\right) \in \mathbf{s}(x1​,x2​,…),y=(y1​,y2​,…)∈s and all α∈F,\alpha \in \mathbb{F},α∈F, define
x+y=(x1+y1,x2+y2,…)α⋅x=(αx1,αx2,…)\begin{aligned} x+y &=\left(x_{1}+y_{1}, x_{2}+y_{2}, \ldots\right) \\ \alpha \cdot x &=\left(\alpha x_{1}, \alpha x_{2}, \ldots\right) \end{aligned} x+yα⋅x​=(x1​+y1​,x2​+y2​,…)=(αx1​,αx2​,…)​
Then s\mathbf{s}s is a linear space over F\mathbb{F}F.

[3] The sequence space ℓ∞.\ell_{\infty} .ℓ∞​. Let ℓ∞=ℓ∞(N)\ell_{\infty}=\ell_{\infty}(\mathbb{N})ℓ∞​=ℓ∞​(N) denote the set of all bounded sequences of real or complex numbers. That is, all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ such that
sup⁡i∈N∣xi∣<∞\sup _{i \in \mathbb{N}}\left|x_{i}\right|<\infty i∈Nsup​∣xi​∣<∞
Define the operations of addition and scalar multiplication pointwise as in [2]. Then ℓ∞\ell_{\infty}ℓ∞​ is a linear space over F\mathbb{F}F.

[4] The sequence space ℓp=ℓp(N),1≤p<∞\ell_{p}=\ell_{p}(\mathbb{N}), \quad 1 \leq p<\inftyℓp​=ℓp​(N),1≤p<∞. Let ℓp\ell_{p}ℓp​ denote the set of all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ of real or complex numbers satisfying the condition
∑i=1∞∣xi∣p<∞\sum_{i=1}^{\infty}\left|x_{i}\right|^{p}<\infty i=1∑∞​∣xi​∣p<∞
Define the operations of addition and scalar multiplication pointwise: For all x=(xn),y=x=\left(x_{n}\right), y=x=(xn​),y= (yn)\left(y_{n}\right)(yn​) in ℓp\ell_{p}ℓp​ and all α∈F,\alpha \in \mathbb{F},α∈F, define
x+y=(x1+y1,x2+y2,…)α⋅x=(αx1,αx2,…)\begin{aligned} x+y &=\left(x_{1}+y_{1}, x_{2}+y_{2}, \ldots\right) \\ \alpha \cdot x &=\left(\alpha x_{1}, \alpha x_{2}, \ldots\right) \end{aligned} x+yα⋅x​=(x1​+y1​,x2​+y2​,…)=(αx1​,αx2​,…)​
[5] The sequence space c=c(N)\mathbf{c}=\mathbf{c}(\mathbb{N})c=c(N). Let c\mathbf{c}c denote the set of all convergent sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ of real or complex numbers. That is, c\mathbf{c}c is the set of all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ such that lim⁡n→∞xn\lim _{n \rightarrow \infty} x_{n}limn→∞​xn​ exists. Define the operations of addition and scalar multiplication pointwise as in [2]. Then c\mathbf{c}c is a linear space over F\mathbb{F}F.

[6] The sequence space c0=c0(N).\mathbf{c}_{0}=\mathbf{c}_{0}(\mathbb{N}) .c0​=c0​(N). Let c0\mathbf{c}_{0}c0​ denote the set of all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ of real or complex numbers which converge to zero. That is, c0\mathbf{c}_{0}c0​ is the space of all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ such that lim⁡n→∞xn=0.\lim _{n \rightarrow \infty} x_{n}=0 .limn→∞​xn​=0. Define the operations of addition and scalar multiplication pointwise as in example (3). Then c0\mathbf{c}_{0}c0​ is a linear space over F\mathbb{F}F.

[7] The sequence space ℓ0=ℓ0(N).\ell_{0}=\ell_{0}(\mathbb{N}) .ℓ0​=ℓ0​(N). Let ℓ0\ell_{0}ℓ0​ denote the set of all sequences x=(xn)1∞x=\left(x_{n}\right)_{1}^{\infty}x=(xn​)1∞​ of real or complex numbers such that xi=0x_{i}=0xi​=0 for all but finitely many indices iii. Define the operations of addition and scalar multiplication pointwise as in example (3). Then ℓ0\ell_{0}ℓ0​ is a linear space over F\mathbb{F}F.

1.2 Subsets of a linear space

Notation

Let XXX be a linear space over F,x∈X\mathbb{F}, x \in XF,x∈X and AAA and BBB subsets of XXX and λ∈F\lambda \in \mathbb{F}λ∈F. We shall denote by
x+A:={x+a:a∈A}A+B:={a+b:a∈A,b∈B}λA:={λa:a∈A}\begin{aligned} x+A &:=\{x+a: a \in A\} \\ A+B &:=\{a+b: a \in A, b \in B\} \\ \lambda A &:=\{\lambda a: a \in A\} \end{aligned} x+AA+BλA​:={x+a:a∈A}:={a+b:a∈A,b∈B}:={λa:a∈A}​

1.3 Subspaces and Convex Sets

Subspaces

A subset MMM of a linear space XXX is called a linear subspace of XXX if
(a) x+y∈Mx+y \in Mx+y∈M for all x,y∈M,x, y \in M,x,y∈M, and
(b) λx∈M\lambda x \in Mλx∈M for all x∈Mx \in Mx∈M and for all λ∈F\lambda \in \mathbb{F}λ∈F.
    Clearly, a subset MMM of a linear space XXX is a linear subspace if and only if M+M⊂MM+M \subset MM+M⊂M and λM⊂M\lambda M \subset MλM⊂M for all λ∈F\lambda \in \mathbb{F}λ∈F.

Then Every linear space XXX has at least two distinguished subspaces: M={0}M=\{0\}M={0} and M=XM=XM=X. These are called the improper subspaces of X.X .X. All other subspaces of XXX are called the proper subspaces.

linear hull

Let KKK be a subset of a linear space X.X .X. The linear hull of K,K,K, denoted by lin⁡(K)\operatorname{lin}(K)lin(K) or span⁡(K),\operatorname{span}(K),span(K), is the intersection of all linear subspaces of XXX that contain KKK.
    The linear hull of KKK is also called the linear subspace of XXX spanned (or generated) by KKK.
    It is easy to check that the intersection of a collection of linear subspaces of XXX is a linear subspace ofX.X .X. It therefore follows that the linear hull of a subset KKK of a linear space XXX is again a linear subspace of X.X .X.
     In fact, the linear hull of a subset KKK of a linear space XXX is the smallest linear subspace of XXX which contains KKK
(实际上，X的子集K的线性包是线性空间X的包含K的最小线性子空间)

proposition

Let KKK be a subset of a linear space X.X .X. Then the linear hull of KKK is the set of all finite linear combinations of elements of K.K .K. That is,
lin⁡(K)={∑j=1nλjxj∣x1,x2,…,xn∈K,λ1,λ2,…,λn∈F,n∈N}\operatorname{lin}(K)=\left\{\sum_{j=1}^{n} \lambda_{j} x_{j} \mid x_{1}, x_{2}, \ldots, x_{n} \in K, \lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} \in \mathbb{F}, n \in \mathbb{N}\right\} lin(K)={j=1∑n​λj​xj​∣x1​,x2​,…,xn​∈K,λ1​,λ2​,…,λn​∈F,n∈N}

Other Definition

[1]A subset KKK of a linear space XXX is said to be linearly independent if every finite subset {x1,x2,…,xn}\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}{x1​,x2​,…,xn​} of KKK is linearly independent.

[2]If {x1,x2,…,xn}\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}{x1​,x2​,…,xn​} is a linearly independent subset of XXX and X=lin⁡{x1,x2,…,xn},\bold {X=\operatorname{lin}\left\{x_{1}, x_{2}, \ldots, x_{n}\right\},}X=lin{x1​,x2​,…,xn​}, then XXX is said to have dimension n.n .n. In this case we say that {x1,x2,…,xn}\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}{x1​,x2​,…,xn​} is a basis for the linear space X.X .X. If a linear space XXX does not have a finite basis, we say that it is infinitedimensional.

Convex Sets

Let KKK be a subset of a linear space X.X .X. We say that
(a) KKK is convex if λx+(1−λ)y∈K\lambda x+(1-\lambda) y \in Kλx+(1−λ)y∈K whenever x,y∈Kx, y \in Kx,y∈K and λ∈[0,1]\lambda \in[0,1]λ∈[0,1]
(b) KKK is balanced if λx∈K\lambda x \in Kλx∈K whenever x∈Kx \in Kx∈K and ∣λ∣≤1|\lambda| \leq 1∣λ∣≤1;
(c）KKK is absolutely convex if KKK is convex and balanced.

Convex hull

Let SSS be a subset of the linear space X.X .X. The convex hull of S,S,S, denoted co⁡(S),\operatorname{co}(S),co(S), is the intersection of all convex sets in XXX which contain SSS.

Since the intersection of convex sets is convex, it follows that co⁡(S)\operatorname{co}(S)co(S) is the smallest convex set which contains SSS.
（实际上，X子集S的凸包是X中包含S最小的凸包）

proposition

Let SSS be a nonempty subset of a linear space X.X .X. Then co⁡(S)\operatorname{co}(S)co(S) is the set of all convex combinations of elements of S.S .S. That is,
co⁡(S)={∑j=1nλjxj∣x1,x2,…,xn∈S,λj≥0∀j=1,2,…,n,∑j=1nλj=1,n∈N}\operatorname{co}(S)=\left\{\sum_{j=1}^{n} \lambda_{j} x_{j} \mid x_{1}, x_{2}, \ldots, x_{n} \in S, \lambda_{j} \geq 0 \forall j=1,2, \ldots, n, \sum_{j=1}^{n} \lambda_{j}=1, n \in \mathbb{N}\right\} co(S)={j=1∑n​λj​xj​∣x1​,x2​,…,xn​∈S,λj​≥0∀j=1,2,…,n,j=1∑n​λj​=1,n∈N}

Remark

[1] KKK is absolutely convex if and only if λx+μy∈K\lambda x+\mu y \in Kλx+μy∈K whenever x,y∈Kx, y \in Kx,y∈K and ∣λ∣+∣μ∣≤1|\lambda|+|\mu| \leq 1∣λ∣+∣μ∣≤1
[2] Every linear subspace is absolutely convex.
(每个线性子空间都是绝对凸的)

1.4 Quotient Space

equivalence relation

1. A relation R\mathcal{R}R on a set XXX is any subset R\mathcal{R}R of the product X×X,X \times X,X×X, i.e., R\mathcal{R}R consists of specific ordered pairs (x,y),(x, y),(x,y), with x∈Xx \in Xx∈X and y∈Xy \in Xy∈X

2. An equivalence relation on XXX is a relation R\mathcal{R}R that satisfies the following properties, where the notation x∼yx \sim yx∼y means that (x,y)∈R(x, y) \in \mathcal{R}(x,y)∈R :
   reflexivity : x∼x\quad x \sim xx∼x for all x∈Xx \in Xx∈X

   symmetry : x∼y\quad x \sim yx∼y implies y∼xy \sim xy∼x

   transitivity : x∼yx \sim yx∼y and y∼zy \sim zy∼z implies x∼zx \sim zx∼z
   Equivalently, (x,x)∈R(x, x) \in \mathcal{R}(x,x)∈R for all x∈X;x \in X ;x∈X; if (x,y)∈R,(x, y) \in \mathcal{R},(x,y)∈R, then (y,x)∈R;(y, x) \in \mathcal{R} ;(y,x)∈R; if (x,y)∈R(x, y) \in \mathcal{R}(x,y)∈R and (y,z)∈R,(y, z) \in \mathcal{R},(y,z)∈R, then (x,z)∈R(x, z) \in \mathcal{R}(x,z)∈R
   （等价关系就是满足上面三个状态的有序状态对（x，y））

Definition

Let MMM be a linear subspace of a linear space XXX over F\mathbb{F}F. For all x,y∈X,x, y \in X,x,y∈X, define
x≡y(modM)⟺x−y∈Mx \equiv y(\bmod M) \Longleftrightarrow x-y \in M x≡y(modM)⟺x−y∈M
≡\equiv≡ defines an equivalence relation on XXX
    then For x∈X,x \in X,x∈X, denote by
[x]:={y∈X:x≡y(modM)}={y∈X:x−y∈M}=x+M[x]:=\{y \in X: x \equiv y(\bmod M)\}=\{y \in X: x-y \in M\}=x+M [x]:={y∈X:x≡y(modM)}={y∈X:x−y∈M}=x+M
the coset of xxx with respect to MMM.
    [商空间定义在介里]The quotient space X/MX / MX/M consists of all the equivalence classes [x][x][x], x∈X.x \in X .x∈X. The quotient space is also called a factor space. moreover，quotient space is also a linear space.
（商空间包含了所有的等价类[x],[x]其实是满足mod规则的一个集合，集合元素属于X。而且mod运算是X作用于它的一个线性空间上的）详情点击介里~

[商空间的维度定义在介里]The codimension of MMM in XXX is defined as the dimension of the quotient space X/M.X / M .X/M. It is denoted by codim⁡(M)=dim⁡(X/M)\operatorname{codim}(M)=\operatorname{dim}(X / M)codim(M)=dim(X/M).
Clearly, if X=M,X=M,X=M, then X/M={0}X / M=\{0\}X/M={0} and so codim⁡(X)=0\operatorname{codim}(X)=0codim(X)=0

Proposition

Let MMM be a linear subspace of a linear space XXX over F.\mathbb{F} .F. For x,y∈Xx, y \in Xx,y∈X and λ∈F,\lambda \in \mathbb{F},λ∈F, define the operations
[x]+[y]=[x+y]and λ⋅[x]=[λ⋅x][x]+[y]=[x+y] \text { and } \lambda \cdot[x]=[\lambda \cdot x] [x]+[y]=[x+y] and λ⋅[x]=[λ⋅x]
Then X/MX / MX/M is a linear space with respect to these operations.

Remark

zero element in the quotient space is the M.

1.5 Direct Sums and Projections

Direct Sums

Let MMM and NNN be linear subspaces of a linear space XXX over F\mathbb{F}F. We say that XXX is a direct sum of MMM and NNN if
X=M+Nand M∩N={0}X=M+N \text { and } M \cap N=\{0\} X=M+N and M∩N={0}
If XXX is a direct sum of MMM and N,N,N, we write X=M⊕N.X=M \oplus N .X=M⊕N. In this case, we say that MMM (resp. NNN ) is an algebraic complement of NNN (resp. MMM ). moreover，for each x,the representation is unique.

projection

Let MMM and NNN be linear subspaces of a linear space XXX over F\mathbb{F}F such that X=M⊕N.X=M \oplus N .X=M⊕N. Define P:X→XP: X \rightarrow XP:X→X by P(x)=m,P(x)=m,P(x)=m, where x=m+n,x=m+n,x=m+n, with m∈Mm \in Mm∈M and n∈N.n \in N .n∈N. Then PPP is an algebraic projection of XXX onto MMM along N.N .N. Moreover M=P(X)M=P(X)M=P(X) and N=(I−P)(X)=ker⁡(P)N=(I-P)(X)=\operatorname{ker}(P)N=(I−P)(X)=ker(P)

Linearity of P\mathbf{P}P : Let x=m1+n1x=m_{1}+n_{1}x=m1​+n1​ and y=m2+n2,y=m_{2}+n_{2},y=m2​+n2​, where m1,m2∈Mm_{1}, m_{2} \in Mm1​,m2​∈M and n1,n2∈N.n_{1}, n_{2} \in N .n1​,n2​∈N. For α∈F\alpha \in \mathbb{F}α∈F
P(αx+y)=P((αm1+m2)+(αn1+n2))=αm1+m2=αPx+PyP(\alpha x+y)=P\left(\left(\alpha m_{1}+m_{2}\right)+\left(\alpha n_{1}+n_{2}\right)\right)=\alpha m_{1}+m_{2}=\alpha P x+P y P(αx+y)=P((αm1​+m2​)+(αn1​+n2​))=αm1​+m2​=αPx+Py
Idempotency of P:\mathbf{P}:P: Since m=m+0,m=m+0,m=m+0, with m∈Mm \in Mm∈M and 0∈N,0 \in N,0∈N, we have that Pm=mP m=mPm=m and hence P2x=Pm=m‾=Px.\overline{P^{2} x=P m=m}=P x .P2x=Pm=m=Px. That is, P2=PP^{2}=PP2=P
Finally, n=x−m=(I−P)x.n=x-m=(I-P) x .n=x−m=(I−P)x. Hence N=(I−P)(X).N=(I-P)(X) .N=(I−P)(X). Also, Px=0P x=0Px=0 if and only if x∈N,x \in N,x∈N, i.e., ker⁡(P)=N\operatorname{ker}(P)=Nker(P)=N

1.6 The Holder and Minkowski Inequalities

conjugate exponents and some Inequalities

Let ppp and qqq be positive real numbers. If 1<p<∞1<p<\infty1<p<∞ and 1p+1q=1,\frac{1}{p}+\frac{1}{q}=1,p1​+q1​=1, or if p=1p=1p=1 and q=∞,q=\infty,q=∞, or if p=∞p=\inftyp=∞ and q=1,q=1,q=1, then we say that ppp and qqq are conjugate exponents.

Lemma (Young’s Inequality). Let ppp and qqq be conjugate exponents, with 1<p,q<∞1<p, q<\infty1<p,q<∞ and α,β≥0.\alpha, \beta \geq 0 .α,β≥0. Then
αβ≤αpp+βqq\alpha \beta \leq \frac{\alpha^{p}}{p}+\frac{\beta^{q}}{q} αβ≤pαp​+qβq​

Holder Inequalities

(Hölder’s Inequality for sequences). Let (xn)∈ℓp\left(x_{n}\right) \in \ell_{p}(xn​)∈ℓp​ and (yn)∈ℓq,\left(y_{n}\right) \in \ell_{q},(yn​)∈ℓq​, where p>1p>1p>1 and 1/p+1/q=11 / p+1 / q=11/p+1/q=1. Then
∑k=1∞∣xkyk∣≤(∑k=1∞∣xk∣p)1p(∑k=1∞∣yk∣q)1q\sum_{k=1}^{\infty}\left|x_{k} y_{k}\right| \leq\left(\sum_{k=1}^{\infty}\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{\infty}\left|y_{k}\right|^{q}\right)^{\frac{1}{q}} k=1∑∞​∣xk​yk​∣≤(k=1∑∞​∣xk​∣p)p1​(k=1∑∞​∣yk​∣q)q1​

Minkowski Inequalities

Theorem (Minkowski’s Inequality for sequences). Let p>1p>1p>1 and (xn)\left(x_{n}\right)(xn​) and (yn)\left(y_{n}\right)(yn​) sequences in ℓp\ell_{p}ℓp​. Then
(∑k=1∞∣xk+yk∣p)1p≤(∑k=1∞∣xk∣p)1p+(∑k=1∞∣yk∣p)1p\left(\sum_{k=1}^{\infty}\left|x_{k}+y_{k}\right|^{p}\right)^{\frac{1}{p}} \leq\left(\sum_{k=1}^{\infty}\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}+\left(\sum_{k=1}^{\infty}\left|y_{k}\right|^{p}\right)^{\frac{1}{p}} (k=1∑∞​∣xk​+yk​∣p)p1​≤(k=1∑∞​∣xk​∣p)p1​+(k=1∑∞​∣yk​∣p)p1​

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