Shuailong Zhu

1 Classical Solutions

1.1 Classifications

Definition 1.1 (Poisson equations with different boundaries)

Dirichlet boundary:

$$− \Delta u = f \text{ in } \Omega, \\ u = g \text{ on } \partial\Omega$$

Neumann boundary:

$$− \Delta u = f \text{ in } \Omega, \\ \partial u = h \text{ on } \partial\Omega$$

If $f=0$, we call it Laplace equation. Classically we restrict the solution $u \in C^2(\Omega) \cap C^0(\overline\Omega)$.

1.2 Uniqueness and Existence

2 Weak Solutions

2.1 Abstract Formulation

Consider the following problem,

$$− \Delta u = f \text{ in } \Omega, u = 0 \text{ on } \partial\Omega \tag{2.1}$$

Definition 2.1 (Weak solution). We say that $u \in H^1_0(\Omega)$ is a weak solution if

$$\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} fv, ∀v \in H^1_0(\Omega)$$

which is called the weak formulation of (2.1). $f\in L^2(\Omega)$ is enough, but $f$ might also be distribution.

Definition 2.2 (Abstract variational problem).

Let $V$ be a Hilbert space,

$$\text{find } u \in V : a(u, v) = F(v), \forall v \in V$$

---

Formulating in the abstract way, let $V=H^1_0 (\Omega)$ , We say that $u \in V$ is a weak solution of (2.1) with $f \in V’$ if

$$a(u, v) = F(v), \forall v \in V.$$

where $a(u, v) = \int_{\Omega} \nabla u \cdot \nabla v, F(v)= \left< f,v\right >$. We can denote $(H^1_0 (\Omega))’$ by $H^{−1 }(\Omega)$.

We can utilize the abstract form with Lax-Milgram theorem in PDEs.

2.2 Applications to PDEs

2.2.1 Homogeneous Poisson equation with Dirichlet boundary.

Lemma 2.1 For any $f \in H^{−1 }(\Omega)$, (2.1) in a bounded domain $\Omega$ has a unique weak solution $u ∈ H^1_0 (\Omega)$ which satisfies $\Vert u\Vert_{H^1_0 (\Omega)} = \Vert f\Vert_{H^{−1}(\Omega)}$.

Proof.

(1) Coercivity of $a$.

Using Poincare inequality,

$$\Vert f\Vert_{L^2(\Omega)} \leq C_P \Vert Df\Vert_{L^2(\Omega)}, ∀f \in H^1_0(\Omega).$$

We can show $\Vert\cdot\Vert_{L^2}$ is equivalent to $\Vert\cdot\Vert_{H^1_0}$. Thus, it is direct to get $\alpha=1$.

2.2.2 Non-homogeneous Poisson equation with Dirichlet boundary.

$$− \Delta u = f \text{ in } \Omega, u = g \text{ on } \partial\Omega \tag{2.2}$$

Lemma 2.2. The space $Rτ \subset L^2 (\partial\Omega)$ a strict subspace, endowed with the norm

$$\Vert g\Vert_{R\tau} = \mathop{inf}\limits_{u\in H^1(\Omega), \tau u=g}\Vert u\Vert _{H^1(\Omega)}.$$

is a Banach space, which is usually denoted by $H^{\frac{1}{2}}(\partial\Omega)$.

Definition 2.3 Let $\Omega \subset \mathbb R^n$ be a bounded domain with $C^1$ boundary. We say that $u \in H^1 (\Omega)$ is a weak solution of (2.2) with $f \in H^{-1}(\Omega)$ and $g \in H^{1/2} (\partial\Omega)$ if it satisfies $\tau u = g$ and

$$\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} fv, \forall v \in H_0^1(\Omega)$$

Let $u=u_0+v_g$ such that $\tau v_g=g$ and $v_g\in H^1(\Omega)$, $u_0\in H^1_0(\Omega)$

$$\int_{\Omega}\nabla u \cdot \nabla v=\int_{\Omega} fv-\int_{\Omega}\nabla v_g \cdot \nabla v, \forall v \in H_0^1(\Omega)$$

Remark. According to Lemma 2.2, it is necessary to have some constraint on $g$.

Lemma 2.3 Let $\Omega \subset \mathbb R^n$ be a bounded domain with $C^1 $ boundary. For any $g \in H^{1/2} (\partial\Omega)$ and $f \in H^{-1} (\Omega)$ there exists a unique weak solution to problem.

2.2.3 Non-homogeneous Generalized Poisson equation with Neumann boundary.

The Problem is

$$− \Delta u +cu= f \text{ in } \Omega, \partial_v u = h \text{ on } \partial\Omega \tag{2.3}$$

Definition 2.4. We say that $u \in H^1 (\Omega)$​ is a weak solution of (2.3) with $f \in L^2 (Ω)$​ and $h \in L^2 (\partial\Omega)$​ if

$$\int_{\Omega}\nabla u \cdot \nabla v+\int_{\Omega} u v=\int_{\Omega} fv+\int_{\partial \Omega} h \tau v,\forall v \in H^1 (\Omega).$$

Remark

(1) Here we must require $c>0$ to ensure coercivity of $a$.

(2) Generally, $f\in H^{-1}(\Omega)$ is enough. Since $\tau$ is also a bounded operator, $h\in H^{-1/2}(\Omega)=(H^{1/2}(\Omega))’$ is also enough.

Then the theorem is similarly stated.

1 $L^p$ Space

1.1 Density

Theorem 1.1 (Density of simple functions) Let $(X,\mathscr{A},\mu)$ be a measure space, and let $p$ satisfy $1 \leq p \leq +\infty$. Then the simple functions in $L^p(X,\mathscr{A},\mu)$ form a dense subspace of $L^p(X,\mathscr{A},\mu)$.

Lemma 1.2 (Approximation of measurable function by simple function) If $f : X \rightarrow [0, \infty]$ is a positive measurable function, then there is a monotone increasing sequence of positive simple functions $\varphi_n : X → [0, \infty)$ with $\varphi_1 ≤ \varphi_2 ≤ · · · ≤ \varphi_n ≤ . . .$ such that $\varphi_n \rightarrow f$ pointwise. If $f$ is bounded, then $\varphi_n \rightarrow f$ uniformly.

Proof of Lemma 1.2 For each $n \in \mathbb N$, we divide the interval $[0, n]$ into $n2^{n}$ subintervals of width $1/2^n$

$$I_{k,n} = (k2^{−n} ,(k + 1)2^{−n}], k = 0, 1, . . . , n2^{n }-1,$$

and let $E_n = (n, ∞]$. Define

$$E_{k,n} = f^{-1} (I_{k,n}), F_n = f ^{−1} (J_n).$$

Let $\varphi_n(x) = n\chi_{E_n} (x) + \sum\limits_{k=0}^{n2^n-1}\dfrac{k}{2^n}\chi_{E_{k,n}(x)}$. Then $\varphi_n(x)\leq\varphi_{n+1}(x)\leq f(x)$. We show pointwise convergence in the following.

(1) For $x$ such that $f(x)< M$ if $x\geq M$, then for $n>M$, $f(x)-\varphi_n(x)<\dfrac{1}{2^n}$.

(2) For $x$ such that $f(x)=+\infty$, $lim_{n\rightarrow \infty}\varphi_n(x)=+\infty$.

Proof of Theorem 1. It is sufficient to prove that we can approximate a positive function $f : X → [0, ∞]$ by simple functions.

First suppose that $f \in L_p (X)$ where $1 \leq p < \infty$. Then, from Theorem 1.1, there is an increasing sequence of simple functions $\{\varphi_n\}$ such that $\varphi_n \rightarrow f$ pointwise. Since $|\varphi_n(x)|\leq |f(x)|$, $\varphi_n\in L_p(X)$. Also, $|f − \varphi_n|^{p} ≤ |f|^{p}$ suggests that $|f − \varphi_n|^{p}$ is integrable. As we have $\varphi_n\rightarrow f$ pointwise, $|f − \varphi_n|^{p}\rightarrow 0$ pointwise. Using dominated convergence theorem,

$$\int|f − \varphi_n|^pd\mu\rightarrow 0$$

Secondly, if $f\in L^{\infty}(X)$, then we may choose a representative of $f$ that is bounded. According to Theorem 1, there is a sequence of simple functions that converges uniformly to $f$, and therefore in $L^{\infty}(X)$.

Theorem 1.3 (Density of continuous functions) The space $C_c(\mathbb R_n)$ is dense in $L^1 (\mathbb R_n)$. (could be extended to $L^p$ for $p\in[1,\infty)$)

Proof (1). Using Borel regularity of Lebesgue measure and Urysohn’s lemma.

Proof (2). Using Lusin’s theorem.

1.2 Completeness

Theorem 1.4 (Riesz–Fischer theorem) $L^p (X)$ is complete for $p\in[1,\infty]$.

Proof. We only prove this for $p\in[1,\infty)$.

(1) Let $f_{n_0}$ be a cauchy sequence in $L^p(X)$. There exists a subsequence $\{f_n\}$ such that

$$\sum\limits_{k=1}^{\infty}||f_{k+1}-f_k||_p<\infty$$

Let

$$g_n=|f_1|+\sum\limits_{k=1}^{n-1}|f_{k+1}-f_k|$$

For each $x\in X$, $\{g_n(x)\} \uparrow$, and thus $g(x)=limg_n(x)$ (in the sense of $[0,\infty]$). By monotone convergence theorem and the fact that $g_n^p$ is a monotone sequence, $\int g^p_nd\mu\rightarrow\int gd\mu$. Since

$$||g_n||_p\leq||f_1||_p+\sum\limits_{k=1}^{\infty}||f_{k+1}-f_k||_p<\infty$$

$g\in L^p(X)$ and $g$ is almost everywhere finite. We also have

$$g_m(x)-g_n(x)\geq |f_m(x)-f_n(x)| \text{ for } m>n \tag{1}$$

(2) Since $g$ is almost everywhere finite and $g(x)=limg_n(x)$, $\mathop{lim}\limits_{m,n\rightarrow\infty} g_m(x)-g_n(x)=0$ holds almost everywhere, and so do $\mathop{lim}\limits_{m,n\rightarrow\infty} f_m(x)-f_n(x)=0$. Thus $\{f_n(x)\}$ forms a cauchy sequence almost everywhere $\Rightarrow$ $f_n(x)$ has finite limit almost everywhere, which is denoted as $f(x)$. Let $m\rightarrow\infty$,

$$|f(x)-f_n(x)|\leq g(x)-g_n(x)\leq g(x), a.e.$$

$|f(x)|\leq f_n(x)+g(x)\Rightarrow f(x)\in L^p(X)$. Also,

$$|f(x)-f_n(x)|^p\leq [g(x)]^p \Rightarrow |f(x)-f_n(x)|^p\in L(X)$$

According to Dominated convergence theorem, $||f(x)-f_n(x)||_p\rightarrow 0$.

2 Sobolev Spaces

2.1 Weak Derivative

Definition 2.1 (Distributional derivative) For $f ∈ L^1_{loc}(\Omega)$ and $\alpha \in \mathbb{N}^n$ , the α-distributional partial derivative $D^{\alpha}f \in \mathscr{D}’ (Ω)$ is defined as

$$\left <D^{\alpha} f, \varphi \right>:= (−1)^{|\alpha|} \int_{\Omega} fD^{\alpha}ϕ\varphi dx, \forall \varphi \in \mathscr D(Ω).$$

Definition 2.2 (Weak derivative) For $f ∈ L^1_{loc}(\Omega)$ and $\alpha \in \mathbb{N}^n$ , we say $f$ has α-weak derivative if its distributional derivative $D^{\alpha}f \in L_{loc}^1(\Omega), i.e. $ there exists $v\in L_{loc}^1(\Omega)$ s.t.

$$\int_{\Omega}v\varphi dx= (−1)^{|\alpha|} \int_ Ω fD^{\alpha}ϕ\varphi dx, , \forall \varphi \in \mathscr D(Ω).$$

Definition 2.3 (Sobolev spaces). For $1 \leq p \leq\infty$, we define,

$$W^{k,p}(\Omega) = \{f \in L^p (\Omega) : D^{\alpha} f \in (\Omega) ,\forall |\alpha| \leq k\}$$

with the norm

$$|| f||_{k,p}= \left(\sum\limits_{|\alpha|\leq k}\int_{\Omega}|D^{\alpha }f | ^ p \right )^{\frac{1}{p}}$$

2.2 Density Results of Different Mollifications

Definition 2.4 (Standard mollification)

Theorem 2.1 Let $f \in W^{k,p}(\Omega), 1 \leq p < \infty$ and $f_{\epsilon} = η ∗ f : \Omega \rightarrow \mathbb {R}$. Then

$$f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } L^p(\Omega)\\ f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } W^{k,p}(K) ,\forall K\subset\Omega$$

The result is the same for mollification from inside.

Proposition 2.2 $C_{0}^{\infty}(\Omega)$ is not dense in $W^{k,p}(\Omega)$.

Proof. Under some conditions like $C^1$ boundary, $\tau_0 u=\mathop{lim}\limits_{k\rightarrow\infty}u_k|_{\partial u}=0$. (Theorem 5.8)

Definition 2.5 Following Proposition 2.3, we denote by $W^{k,p}_0(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$ in $W^{k,p}(\Omega)$.

Theorem 2.3 (Meyers-Serrin’s theorem) $C^{\infty}(\Omega)\cap W^{k,p}(\Omega)$ is dense in $W^{k,p}(\Omega)$.

Definition 2.6 (translated mollification)

Theorem 2.4 Let $\Omega = \mathbb R^n_ {+}$ and $1 \leq p < \infty$. Let $\hat{f_{\epsilon}}$ be translated mollification. Then

$$\hat f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } L^p(\Omega)\\ \hat f_{\epsilon} \mathop{\longrightarrow}\limits_{}^{\epsilon\rightarrow 0} f \text{ in } W^{k,p}(\Omega)$$

Also, $\hat f_{\epsilon}\in C_0^{\infty} (\overline\Omega)$.

Theorem 2.5 If $\Omega$ satisfies the segment condition, then $C^{\infty} (\overline\Omega)$ is dense in $W^{k,p}(\Omega)$, for $1 \leq p < \infty$. [Theorem 3 in Section 5.3.3 in Evan’s book]

2.3 Traces

Deal with the boundary: Since $L^p$ space has no restrictions for boundary (null set), it is meaningless to discuss the solution in $L^p$ space. Thus, we turn to $W^{k,p}(\Omega)$. The way to proceed is the following: since $C^{\infty}(\overline\Omega)$ is dense in $W^{k,p}(\Omega)$ [Theorem 2.5], given $f \in W^{k,p}(\Omega)$ we can find a sequence $f_k \in C^{\infty}(Ω)$ s.t.

$$f_k\rightarrow f \text{ in }W^{k,p}(\Omega)$$

For each $f_k$, the trace $f_k|\partial\Omega$ is uniquely defined, so that we define the “trace of $f$ on ∂Ω” as $\mathop{lim}\limits_{k\rightarrow \infty} f_k|\partial\Omega$. The crucial question is whether such limit exists and with respect to which topology.

Theorem 2.5 (Trace theorem in $\mathbb R^n_+$). There exists a bounded linear operator $τ_0 : W^{1,p}(\mathbb R^n_+) \rightarrow L^p (\partial \mathbb R^n_+), 1 \leq p < \infty$, s.t. $\tau_0u = u|\partial {\mathbb R_n^+}$ if $u \in W^{1,p}(\mathbb R^n_+) ∩ C_0 (\overline{\mathbb R^+_n})$.

Theorem 2.6 (Trace theorem in $\Omega$ ). Let $\Omega$ be a domain with bounded boundary $\partial\Omega$ of class $C^1$. There exists a bounded linear operator $τ_0 : W^{1,p}(\Omega) \rightarrow L^p (\partial\Omega), 1 \leq p < \infty$, s.t. $\tau_0u = u|\partial {\Omega}$ if $u \in W^{1,p}(\Omega) ∩ C_0 (\overline{\Omega})$.

There are several corollaries:

• Integration by part.

• Poincare inequality: Let $\Omega \subset \mathbb R^n$ be a bounded domain. Then there exists a constant $C_P>0$ such that

  $$\Vert f\Vert_{L^p(\Omega)} \leq C_P \Vert Df\Vert_{L^p(\Omega)}, ∀f \in W^{1,p}_0(\Omega).$$

1 Dual Space

1.1 Dual Space

Definition 1.1 (Dual space) Let $X$ be a normed space. $X^{\ast}=\mathcal{L}(X,K)$ is called the dual space of $X$. Since $K$ is complete, $X^{\ast}$ is also a Banach space.

Definition 1.2 (Bidual space) We can define another intermediate mapping $x^{\ast\ast}: X^{\ast}\rightarrow R$ by $x^{\ast\ast}(f)=f(x)$. The space $X^{\ast\ast}=\mathcal{L}(X^{\ast},K)$ is called Bidual space, which is also a Banach space.

Definition 1.3 (Canonical mapping/embedding) We can define a mapping $J:X\rightarrow X^{\ast\ast}$ where $Jx(f)=f(x)$. $J$ is a linear isometry.

Proof. Since $x^{\ast\ast}(f)=|f(x)|\leq||x||||f||$, $||x^{\ast\ast}||\leq ||x||$. Due to Hahn-Banach theorem, there exists $f_0\in E^{\ast}$ such that $\Vert f_0\Vert=1$ and $|f_0(x)|=||x||$, and thus $||x^{\ast\ast}||\geq||x||$. Therefore, $||x^{\ast\ast}||=||x||$.

Definition 1.4 (Reflexive) In the sense of isometry, $X$ is a subspace of $X^{\ast\ast}$. $X$ is said to be reflexive if the canonical embedding $J$ into its bidual $X^{\ast\ast}$ is surjective.

1.2 Dual Space of Hilbert Space

Theorem 1.1 (Projection theorem) Let $H$ be a Hilbert space and $H_0 \subset H$ a closed subspace. Then for every $x \in H$ there exist uniquely $y \in H_0$ and $z \in H_0^{\perp}$ s.t. $x = y + z$.

Theorem 1.2 (Riesz’s representation theorem) Let $H$ be a Hilbert space $f$ be a bounded linear functional on $H$, then there exists one and only one $h\in H$ such that

$$f(x)=(x,h), ||f||_{H^\ast}=||h||_{H}$$

The Riesz representation theorem suggests that the mapping $H\rightarrow H^{\ast}$ is surjective, and hence an isometric isomorphism. It is not hard to see that $H^{\ast}$ is also a Hilbert space. Repeat the argument for $H^*$ and mapping $H^{\ast}\rightarrow H^{\ast\ast}$. We can prove that a Hilbert space is reflexive.

Corollary 1.3 (Lax-Milgram) Let $V$ be a Hilbert space, $a : H \times H → \mathbb R$ a bilinear, bounded and coercive form on $H$ . Then, for any $f \in H’$ the abstract problem

$$\text{find } u \in H : a(u, v) = f(v), \forall v \in H$$

has a unique solution $u \in V$ such that $\Vert u\Vert_H ≤\Vert f\Vert_{H’}/\alpha$ where $\alpha$ is the coercivity constant of the bilinear form.

1.3 Examples of Dual Spaces

Distribution.

2 Adjoint Operators

Definition 2.1 (Adjoint Operator) Let $X$ and $Y$ be normed linear spaces and $T\in \mathcal{L}(X,Y)$. The adjoint operator of $T$, denoted by $T^{\ast}$ , is the operator $T^{\ast}:Y^{\ast}\rightarrow X^{\ast}$ defined by

$$(T^{\ast}y^{\ast})(x)=y^{\ast}(Tx) \text{ for all } y^{\ast}\in Y^{\ast}, x\in X^{\ast}$$

Theorem 2.1 Let $X$ and $Y$ be normed linear spaces and $T\in \mathcal{L}(X,Y)$.

(a) $T^{\ast}$ is bounded linear operator on $Y^{\ast}$.

(b) $||T||=||T^{\ast}||$.

3 Weak Topology

3.1 Type of Convergence

Definition 3.1 (Types of convergence for sequence)

Let $X$ be normed linear space, $x,\{x_n\}\in X$.

(1) Strong Convergence: $||x_n-x||\rightarrow 0$.

(2) Weak convergence: If $\forall$ $f\in X^{\ast}$, $f(x_n)\rightarrow f(x)$, then we call $x_n\rightharpoonup x$ in weak topology.

(3) Weak-$\ast$ convergence [Concepts for dual space of X]. Let $X$ be a normed space and $X^{\ast}$ be its dual space. We define $f_n\mathop{\rightarrow}\limits_{}^{w^{\ast}} f$ if $f_n(x)\rightarrow f(x)$, for all $x\in X$.

Note: We can define similar convergences for functionals [strong convergence and weak convergence, weak-$\ast$ convergence is originally defined on $X^{\ast}$]. It is not the case for ‘convergence for operators’. For example, when we define weak convergence for $X^{\ast}$: we say $f_n\rightharpoonup f$, if $X^{\ast\ast}(f_n)\rightarrow X^{\ast\ast}(f)$ for all $x^{\ast \ast}\in X^{**}$. Proposition 3.4 and its proof are a very good explanation for this.

Proposition 3.1 (Sufficient and necessary condition for weak convergence) $x_n\rightharpoonup x$ iff

(1) $\{||x_n||\}$ is bounded.

(2) $\exists$ a dense subset $M\subseteq X^{\ast}$ such that for any $f\in M$, $f(x_n)\rightarrow f(x)$.

Proposition 3.2

If $X$ is finite dimensional, then $\text{strong convergence}\iff \text{weak convergence}$.

Proposition 3.3 (Weak-$\ast$ convergence) Let $X$ be Banach space and $f,\{f_n\}\in X^{\ast}$. Then $f_n\mathop{\rightarrow}\limits_{}^{w^{\ast}} f$ iff

(1) $\{||f_n||\}$ is bounded.

(2) $\exists$ a dense subset $M\subseteq X$ such that for any $x\in M$, $f(x_n)\rightarrow f(x)$.

Definition 3.2 (Types of convergence for operators)

Let $X,Y$ be normed linear spaces, $T,\{T_n\}\in \mathcal L(X,Y)$.

(1) Uniform Convergence.

(2) Strong Convergence: If $\forall$ $x\in X$, $T_nx$ convergences strongly to $Tx$, then we call $T_n\rightarrow T$.

(3) Weak Convergence: If $\forall$ $x\in X$, $T_nx$ convergences weakly to $Tx$ in $Y$, then we call $T_n\rightharpoonup T$.

Let $X,Y$ be normed spaces and $T\in \mathcal L(X,Y)$. The mapping $GT=T^{\ast}$ is a distance-preseved mapping and $Im(G)$ is isometric isomorphism wrt $\mathcal L(X,Y)$.

Proposition 3.4 (Proposition 5.13 in [2], Weak Convergence on $X^{}$ and weak-$$ convergence) Let $X$ be normed spaces and $X^{\ast}$ be the dual space, then

$$\text{weak convergence} \Rightarrow \text{weak-}{\ast} \text{convergence}$$

in $X^{\ast}$. If $X$ is reflexive,

$$\text{weak convergence} \ on \ X^* \iff\text{weak-}{\ast} \text{convergence}$$

3.2 Separability

Theorem 3.5 (Banach-Alaoglu Theorem, Theorem 5.5.7 in [1]). Let $X$ be a normed linear space. Then the closed unit ball in $X$ is weak-$*$ compact.

Proof. We can turn to [1] and link here.

Proposition 3.6 (Sequential Banach-Alaoglu Theorem). If $X$ is a separable normed space, then any bounded sequence $\{f_n\}\in X^{\ast}$ admits a weakly-* convergent subsequence. This is equivalent to Helly selection Theorem. (We cannot treat it as weaker Version of Banach-Alaoglu Theorem.)

Proposition 3.7. Let $X$ be a normed space, then $X^{\ast}$ is separable $\Rightarrow$ $X$ is separable.

3.3 Weak Convergence in $L^2$

Proposition 3.8 (Weak Convergence on $L^p$, Example 4 in Section 5.3 [2]). In $L^p[a,b]$, $x_n\rightharpoonup x_0$ iff:

(1) For every $t\in[a,b]$, we have

$$\int_{a}^t x_n(s)ds\rightarrow \int_a^t x_0(s)ds$$

(2) $(\Vert x_n\Vert )$ is bounded.

Proof. It relies on Proposition 3.1 in this note.

Proposition 3.9 (Weak Convergence on $L^2$ or generally a hilbert space $H$) This link provieds a theorem: Every bounded sequence in Hilbert space admits a weakly convergent sequence.

4 Finite Dimentional Normed Space

A small supplement.

Definition 4.1 (Linear isometry and homeomorphism in normed spaces) In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. Linear isometry is often respect with normed vector spaces. Homeomorphism represents the topological isomorphism.

Proposition 4.1 (Several Properties) I met them in my “Smooth Manifolds” course.

• All norms on finite dimensional normed space $X$ are equivalent. [1, Theorem 2.6.1]
• Every finite dimensional normed space is linear isomorphic (线性同构，一定意义的代数上) and homeomorphic (同胚，拓扑上) to $\mathbb{R}^n$. [2, Theorem 2.1]

Proposition 4.2 (More properties about the finite dim normed spaces)

• Every finite dimensional normed space is complete. Heine-Borel Theorem could be generalized to finite dimensional normed space.
• A normed space $X$ is finite dimensional iff $X$ is locally compact, i.e. every point $x$ in $X$ has a compact neighborhood (through Riesz’s Lemma). [2, Theorem 2.2]
• A normed space $X$ is finite-dimensional iff its closed unit ball is compact. [1, Theorem 2.6.4]

References

[1] Functional Analysis Notes, by Mr. Andrew Pinchuck, https://uomustansiriyah.edu.iq/media/lectures/9/9_2017_09_30!12_03_54_PM.pdf

[2] A Sketch of Real-Value Function and Functional Analysis, by Wang Shengwang, Zheng Weixing.

[Some Hexo Debug Experience: modify inline.js: https://myblackboxrecorder.com/use-math-in-hexo/]

1 Bounded Linear Operator and the Corresponding Space

Theorem 1.1 Let and be normed spaces. is a linear operator, then the following properties are equivalent:

(1) is continuous (in the sense of continuity for operators in metric spaces).

(2) is continuous at .

(3) is bounded.

Theorem 2.2 (The space of bounded linear operators) Let and be normed spaces. is a normed space with . If is Banach space, then is Banach space.

2 Baire Category Theorem and its Corollaries

2.1 Baire Category Theorem

Definition 2.1 (Nowhere dense set): Let a metric space, we say is nowhere dense if . (For any open set , .)

Theorem 2.1 (Baire category theorem): Any complete metric space is not mearge, i.e. can not be written as a countable union of nowhere dense sets.

We use this theorem in the following way: Let be nonempty complete metric space. Let be a sequence of closed subsets such that . Then there exists some such that

$$Int (X_{n_0}) \neq \varnothing$$

2.2 Four Corollaries

Definition 2.2 (Open mapping) We say is an open mapping if is open in whenever is open.

Theorem 2.2 (Open mapping theorem). Let and be Banach spaces and suppose that . If maps onto , then is an open mapping.

Corollary 2.3 (Bounded inverse theorem) Let and be Banach spaces and suppose that . If is bijective, then .

Definition 2.3 (Closed operator) Let and be normed spaces. Let defined on subspace of , be a linear operator.

(1) Assume that the graph , is closed in with norm (also a normed space), then is a closed operator.

(2) For any , if and (), then and is a closed operator.

Theorem 2.4 (Closed graph theorem) Let and be Banach spaces and a closed linear operator. Then is bounded (continuous).

Theorem 2.5 (Uniform boundedness theorem) Let be a Banach space, a normed linear space and let be subset of such that for any , such that,

$$\mathop{\Vert Tx\Vert}\limits_{T\in \mathcal F}<M_x$$

then there exists $M$ such that $\mathop{sup}\limits_{T\in \mathcal{F}}\Vert T\Vert<M$.

Corollary 2.6 Let $X$ be a Banach space, $Y$ be a normed space, and $T,\{T_n\}\in\mathcal L(X,Y)$. Assume $X_0\subseteq X$ is a dense subset. Then for any $x\in X$, $\mathop{lim}\limits_{n\rightarrow \infty}T_nx=Tx$ iff

$$\{||T_n||\} \text{ is bounded}; \mathop{lim}\limits_{n\rightarrow \infty}T_ny=Ty ,\forall y\in X_0$$

Proof See [2] Corollary 2.25.

Corollary 2.7 Let be Banach space, then is complete with respect to strong operator topology.

Proof. See [1] Corollary 6.2.3.

3 Hahn-Banach Theorem and its Consequences

3.1 Hahn-Banach Theorem

Theorem 3.1 (Zorn’s lemma) Let be a partially ordered set. If each totally ordered subset of has an upper bound, then has a maximal element.

Zorn’s Lemma is equivalent to Axiom of Choice: if is any collection of non-empty sets, then also is non-empty.

Definition 3.1 (Sublinear functional) Let be a linear space. The Mapping is called sublinear functional if

(1)

(2) .

Theorem 3.2 (Hahn-Banach theorem) be a proper linear subspace of a real linear space . Suppose that a sublinear functional defined on X, and a linear functional defined on such that for all . Then can be extended to a linear functional defined on such that for .

Proof. see [1].

3.2 Consequences of Hahn-Banach Theorem

Corollary 3.3 Let be a linear subspace. If is a bounded linear functional, then there exists that extends and such that

$$\Vert F\Vert_{X^*}=\Vert f\Vert_{X_0^*}$$

Corollary 3.4 Let be a linear subspace of a normed linear space and such that

$$d(x,x_0)=\mathop{inf}_{x_0\in X_0}||x-x_0||=d$$

then there exists such that,

$$||x^*||_{X^*}=1,x^*(x)=d \\x^*(x_0)=0,\forall x_0\in X_0$$

Corollary 3.5 Let be a normed linear space and , then there exists such that and .

Reference

[1] FUNCTIONAL ANALYSIS NOTES by Mr. Andrew Pinchuck, https://uomustansiriyah.edu.iq/media/lectures/9/9_2017_09_30!12_03_54_PM.pdf

[2] Functional Analysis Notes by He Jiazhi.