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2021/09/29/What-is-an-RKHS-1/index.html

@@ -87,7 +87,7 @@
 <meta property="og:description" content="RKHS随处可见，本系列仅仅为Dino Sejdinovic, Arthur Gretto的What is an RKHS?整理(个人解读+补充)。 第一部分介绍一些必要的泛函的概念。本文中的希尔伯特空间定义使用实数和复数，但是举例仅讨论实数。">
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 <meta property="article:published_time" content="2021-09-29T11:22:37.000Z">
-<meta property="article:modified_time" content="2025-02-05T11:58:28.237Z">
+<meta property="article:modified_time" content="2025-02-05T12:06:10.318Z">
 <meta property="article:author" content="Silven Huang">
 <meta property="article:tag" content="functional">
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@@ -340,14 +340,16 @@ <h1 class="post-title" itemprop="name headline">What is an RKHS [1]</h1>
 <span id="more"></span></p>
 <h2 id="some-functional-analysis-泛函基础">Some functional analysis
 泛函基础</h2>
-<p>介绍基础泛函概念主要是为了得到两个有用的结论: &gt;1)
-在Banach空间上的线性算子是连续的(continus)<span
-class="math inline">\(\iff\)</span>这个算子是有界的(bounded) &gt;2)
-所有定义在Banach空间上的连续线性泛函，都来源于内积。(Riesz
-representation theorem，里斯表示定理)</p>
+<p>介绍基础泛函概念主要是为了得到两个有用的结论:</p>
+<ol type="1">
+<li>在Banach空间上的线性算子是连续的(continus)<span
+class="math inline">\(\iff\)</span>这个算子是有界的(bounded)</li>
+<li>所有定义在Banach空间上的连续线性泛函，都来源于内积。(Riesz
+representation theorem，里斯表示定理)</li>
+</ol>
 <p>这两个结论将帮助我们学习RKHS的性质</p>
-<h3 id="vector-space-linear-space-向量空间线性空间">Vector Space (Linear
-Space) 向量空间(线性空间)</h3>
+<h3 id="vector-space-linear-space-向量空间-线性空间">Vector Space
+(Linear Space) 向量空间 (线性空间)</h3>
 <p>向量空间<span
 class="math inline">\(\mathcal{V}\)</span>是一个在<strong>vector
 addition</strong>和<strong>scalar
@@ -361,38 +363,54 @@ <h3 id="vector-space-linear-space-向量空间线性空间">Vector Space (Linear
 <p><span class="math inline">\(\mathcal{V}\)</span>要成为在<span
 class="math inline">\(\mathcal{F}\)</span>上的向量空间，<span
 class="math inline">\(\forall X,Y,Z\in \mathcal{V}\)</span>，<span
-class="math inline">\(\forall r,s\in \mathcal{F}\)</span>，必须满足： -
-Commutativity <span class="math display">\[X+Y=Y+X\]</span> -
-Associativity of vector addition <span
-class="math display">\[(X+Y)+Z=X+(Y+Z)\]</span> - Additive identivity
-<span class="math display">\[\forall X, 0+X=X+0=X\]</span> - Existance
-of additive inverse <span class="math display">\[\forall X, \exists -X
-\space s.t.\space X+(-X)=0\]</span> - Associativity of scalar
-mutiplication <span class="math display">\[r(sX)=(rs)X\]</span> -
-Distributivity of scalar sums <span
-class="math display">\[(r+s)X=rX+sX\]</span> - Distributivity of vector
-sums <span class="math display">\[r(X+Y)=rX+rY\]</span> - Scalar
-multiplication identity <span class="math display">\[1X=X\]</span></p>
+class="math inline">\(\forall r,s\in
+\mathcal{F}\)</span>，必须满足：</p>
+<ul>
+<li>Commutativity <span class="math display">\[X+Y=Y+X\]</span></li>
+<li>Associativity of vector addition <span
+class="math display">\[(X+Y)+Z=X+(Y+Z)\]</span></li>
+<li>Additive identivity <span class="math display">\[\forall X,
+0+X=X+0=X\]</span></li>
+<li>Existance of additive inverse <span class="math display">\[\forall
+X, \exists -X \space s.t.\space X+(-X)=0\]</span></li>
+<li>Associativity of scalar mutiplication <span
+class="math display">\[r(sX)=(rs)X\]</span></li>
+<li>Distributivity of scalar sums <span
+class="math display">\[(r+s)X=rX+sX\]</span></li>
+<li>Distributivity of vector sums <span
+class="math display">\[r(X+Y)=rX+rY\]</span></li>
+<li>Scalar multiplication identity <span
+class="math display">\[1X=X\]</span></li>
+</ul>
 <p>从固定集合<span
 class="math inline">\(\Omega\)</span>到域F的连续实函数集合<span
 class="math inline">\(\mathcal{f}:\Omega\rightarrow\mathcal{F}\)</span>也可以构成向量空间，<span
 class="math inline">\((f+g)(w)=f(w)+g(w),(c\cdot f)(x)=c\cdot
-f(x)\)</span>。
-&gt;函数空间是一个拓扑向量空间，只不过其中"点"是函数(无限维向量)。
-&gt;齐次微分方程的解就是一个函数空间。所有实函数构成的空间为函数空间<span
-class="math inline">\(\mathbb{F}\)</span>。 &gt;所有<span
-class="math inline">\(n\)</span>次多项式的集合<span
+f(x)\)</span>。</p>
+<blockquote>
+<p>函数空间是一个拓扑向量空间，只不过其中"点"是函数(无限维向量)。</p>
+</blockquote>
+<blockquote>
+<p>齐次微分方程的解就是一个函数空间。所有实函数构成的空间为函数空间<span
+class="math inline">\(\mathbb{F}\)</span>。</p>
+</blockquote>
+<blockquote>
+<p>所有<span class="math inline">\(n\)</span>次多项式的集合<span
 class="math inline">\(\mathbb{P}_n:a_0+a_1x^2+...+a_nx^n\)</span>是<span
 class="math inline">\(\mathbb{F}\)</span>的子空间。</p>
+</blockquote>
 <h3 id="metric-space-度量空间距离空间">Metric Space
 度量空间(距离空间)</h3>
 <p>设<span
 class="math inline">\(\mathcal{X}\)</span>为一个<strong>集合</strong>，一个映射<span
 class="math inline">\(d:\mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}\)</span>。<span
-class="math inline">\(\forall x,y,z\in \mathcal{X}\)</span>，有 - <span
-class="math inline">\(d(x,y)\ge 0,d(x,y)=0\text{ iff } x=y\)</span> -
-<span class="math inline">\(d(x,y)=d(y,x)\)</span> - <span
-class="math inline">\(d(x,z)\leq d(x,y)+d(y,z)\)</span></p>
+class="math inline">\(\forall x,y,z\in \mathcal{X}\)</span>，有</p>
+<ul>
+<li><span class="math inline">\(d(x,y)\ge 0,d(x,y)=0\text{ iff }
+x=y\)</span></li>
+<li><span class="math inline">\(d(x,y)=d(y,x)\)</span></li>
+<li><span class="math inline">\(d(x,z)\leq d(x,y)+d(y,z)\)</span></li>
+</ul>
 <p>那么<span class="math inline">\(d\)</span>是<span
 class="math inline">\(\mathcal{X}\)</span>的一个度量。称偶对<span
 class="math inline">\((\mathcal{X},d)\)</span>是一个度量空间。称<span
@@ -427,26 +445,30 @@ <h3 id="norm-范数">Norm 范数</h3>
 \parallel_\mathcal{F}\leq\parallel f \parallel_\mathcal{F}+\parallel g
 \parallel_\mathcal{F},\forall f,g\in\mathcal{F}\]</span></li>
 </ul>
-<p>赋范空间的所有元素必须有有限的范数。如果有一个元素有无限的范数，那它就不在这个空间里。
-&gt;经典的赋范空间：<span class="math inline">\(L^p\)</span>，<span
+<p>赋范空间的所有元素必须有有限的范数。如果有一个元素有无限的范数，那它就不在这个空间里。</p>
+<blockquote>
+<p>经典的赋范空间：<span class="math inline">\(L^p\)</span>，<span
 class="math inline">\(L^\infty\)</span>，<span
 class="math inline">\(l^p\)</span>，<span
 class="math inline">\(c\)</span>(<span
 class="math inline">\(c_0\)</span>)，<span
 class="math inline">\(V[a,b]\)</span>(<span
 class="math inline">\(V_0[a,b]\)</span>)</p>
+</blockquote>
 <p>范数可以诱导一种在<span
 class="math inline">\(\mathcal{F}\)</span>上的度量：<span
 class="math inline">\(\mathcal{F}:d(f,g)=\parallel f-g
 \parallel_\mathcal{F}\)</span>。说明<span
-class="math inline">\(\mathcal{F}\)</span>具有某种特定拓扑结构，可以让我们研究连续性和收敛性。
-&gt;以绝对值<span
+class="math inline">\(\mathcal{F}\)</span>具有某种特定拓扑结构，可以让我们研究连续性和收敛性。</p>
+<blockquote>
+<p>以绝对值<span
 class="math inline">\(|\cdot|\)</span>为范数的有理数集合<span
 class="math inline">\(\mathbb{Q}\)</span>是一个在其自身上的赋范向量空间，
 序列<span class="math inline">\(1, 1.4, 1.41, 1.414, 1.4142,
 ...\)</span>是一个<span
 class="math inline">\(\mathbb{Q}\)</span>里的不收敛的柯西序列（因为<span
 class="math inline">\(\sqrt{2}\notin\mathbb{Q}\)</span>）</p>
+</blockquote>
 <h3 id="compelete-space-完备空间">Compelete space 完备空间</h3>
 <p>如果<span
 class="math inline">\(\mathcal{X}\)</span>中的所有柯西序列都收敛(有极限，且极限在<span
@@ -467,14 +489,18 @@ <h3 id="inner-product-内积">Inner product 内积</h3>
 }\langle f,f\rangle_\mathcal{F}=0 \text{ iff }f=0\)</span></p>
 <p>可以从内积诱导一个范数<span class="math inline">\(\parallel
 f\parallel_\mathcal{F}=\langle f,f\rangle_\mathcal{F}^{1/2}\)</span></p>
-<p>内积与范数有一些实用的性质: - Cauchy-Schwarz inequality <span
-class="math display">\[|\langle f,g\rangle|\leq\parallel
-f\parallel\cdot\parallel g\parallel\]</span> - the parallelogram law
-<span class="math display">\[\parallel f+g\parallel^2+\parallel
-f-g\parallel^2=2\parallel f\parallel^2+2\parallel g\parallel^2\]</span>
-- the polarization identity(real) <span class="math display">\[4\langle
-f,g\rangle=\parallel f+g\parallel^2-\parallel
-f-g\parallel^2\]</span></p>
+<p>内积与范数有一些实用的性质:</p>
+<ul>
+<li>Cauchy-Schwarz inequality <span class="math display">\[|\langle
+f,g\rangle|\leq\parallel f\parallel\cdot\parallel
+g\parallel\]</span></li>
+<li>the parallelogram law <span class="math display">\[\parallel
+f+g\parallel^2+\parallel f-g\parallel^2=2\parallel
+f\parallel^2+2\parallel g\parallel^2\]</span></li>
+<li>the polarization identity(real) <span
+class="math display">\[4\langle f,g\rangle=\parallel
+f+g\parallel^2-\parallel f-g\parallel^2\]</span></li>
+</ul>
 <blockquote>
 <p>空间的关系：内积空间<span
 class="math inline">\(\subset\)</span>赋范空间<span
@@ -744,8 +770,8 @@ <h3 id="hilbert-space-isomorphism-希尔伯特空间的同构性">Hilbert space
 
 
               <div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#some-functional-analysis-%E6%B3%9B%E5%87%BD%E5%9F%BA%E7%A1%80"><span class="nav-number">1.</span> <span class="nav-text">Some functional analysis
-泛函基础</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#vector-space-linear-space-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4"><span class="nav-number">1.1.</span> <span class="nav-text">Vector Space (Linear
-Space) 向量空间(线性空间)</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#metric-space-%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4%E8%B7%9D%E7%A6%BB%E7%A9%BA%E9%97%B4"><span class="nav-number">1.2.</span> <span class="nav-text">Metric Space
+泛函基础</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#vector-space-linear-space-%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4"><span class="nav-number">1.1.</span> <span class="nav-text">Vector Space
+(Linear Space) 向量空间 (线性空间)</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#metric-space-%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4%E8%B7%9D%E7%A6%BB%E7%A9%BA%E9%97%B4"><span class="nav-number">1.2.</span> <span class="nav-text">Metric Space
 度量空间(距离空间)</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#convergent-sequence-%E6%94%B6%E6%95%9B%E5%BA%8F%E5%88%97"><span class="nav-number">1.3.</span> <span class="nav-text">Convergent sequence 收敛序列</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#cauchy-sequence-%E6%9F%AF%E8%A5%BF%E5%BA%8F%E5%88%97"><span class="nav-number">1.4.</span> <span class="nav-text">Cauchy sequence 柯西序列</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#norm-%E8%8C%83%E6%95%B0"><span class="nav-number">1.5.</span> <span class="nav-text">Norm 范数</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#compelete-space-%E5%AE%8C%E5%A4%87%E7%A9%BA%E9%97%B4"><span class="nav-number">1.6.</span> <span class="nav-text">Compelete space 完备空间</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#banach-space-%E5%B7%B4%E6%8B%BF%E8%B5%AB%E7%A9%BA%E9%97%B4"><span class="nav-number">1.7.</span> <span class="nav-text">Banach space 巴拿赫空间</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#inner-product-%E5%86%85%E7%A7%AF"><span class="nav-number">1.8.</span> <span class="nav-text">Inner product 内积</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#hilbert-space-%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%A9%BA%E9%97%B4"><span class="nav-number">1.9.</span> <span class="nav-text">Hilbert space 希尔伯特空间</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#boundedcontinuous-linear-operators-%E6%9C%89%E7%95%8C%E8%BF%9E%E7%BB%AD%E7%BA%BF%E6%80%A7%E7%AE%97%E5%AD%90"><span class="nav-number">2.</span> <span class="nav-text">Bounded&#x2F;Continuous
 linear Operators 有界&#x2F;连续线性算子</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#linear-operator-%E7%BA%BF%E6%80%A7%E7%AE%97%E5%AD%90"><span class="nav-number">2.1.</span> <span class="nav-text">Linear operator 线性算子</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#continuity-%E8%BF%9E%E7%BB%AD%E6%80%A7"><span class="nav-number">2.2.</span> <span class="nav-text">Continuity 连续性</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#lipschitz-continuity-%E5%88%A9%E6%99%AE%E5%B8%8C%E8%8C%A8%E8%BF%9E%E7%BB%AD"><span class="nav-number">2.3.</span> <span class="nav-text">Lipschitz continuity
 利普希茨连续</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#operator-norm-%E7%AE%97%E5%AD%90%E8%8C%83%E6%95%B0"><span class="nav-number">2.4.</span> <span class="nav-text">Operator norm 算子范数</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#algebraic-dual-%E4%BB%A3%E6%95%B0%E5%AF%B9%E5%81%B6"><span class="nav-number">2.5.</span> <span class="nav-text">Algebraic dual 代数对偶</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#topological-dual-%E6%8B%93%E6%89%91%E5%AF%B9%E5%81%B6"><span class="nav-number">2.6.</span> <span class="nav-text">Topological dual 拓扑对偶</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#riesz-representation-%E9%87%8C%E6%96%AF%E8%A1%A8%E7%A4%BA%E5%AE%9A%E7%90%86"><span class="nav-number">2.7.</span> <span class="nav-text">Riesz representation

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@@ -462,8 +462,8 @@ <h2 id="cvae-条件变分自编码器">CVAE (条件变分自编码器)</h2>
 
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