Functional analysis notes

• Metric Spaces (distance)
• Vector Spaces
• Normed Spaces (distance + length)
• Inner Product Spaces (distance, length, angle)
• Operators

Metric Spaces (distance)

• a set $X$
• Metric $d:X\times X\rightarrow [0, \infty)$
  • $d(x,y) = 0 \Leftrightarrow x=y$
  • $d(x,y) = d(y,x)$
  • $d(x,y) \le d(x,z)+d(y,z)$
• Open ball: $B_\epsilon(x) = \{y\in X | d(x,y)< \epsilon\}$
• Boundary points $\partial A$: $x\in X$ is called a boundary point for $A\subseteq X$ if for all $\epsilon>0$: $B_\epsilon(x)\cap A\ne \emptyset$ and $B_\epsilon(x)\cap A^C\ne \emptyset$
• Open sets: if you are inside the set $A$, you will never see the boundary of the set
  • $A\subseteq X$ is called open if for each $x\in A$ there is an open ball with $B_\epsilon(x)\subseteq A$
  • $A$ open $\Leftrightarrow A\cap \partial A = \emptyset$
• Closed sets: if $A^C$ is open.
  • $A$ closed $\Leftrightarrow A\cap \partial A = \partial A$
• Closure: $\hat{A}:= A\cup \partial A$
• Cauchy Sequences: Let $(X,d)$ be a metric space. A sequence $(x_n)\subseteq X$ is called Cauchy Sequence if $\forall \epsilon>0, \exists N\in \mathbb{N}, \forall n, m\ge N: d(x_n,x_m)<\epsilon$.
• Completeness: $(X, d)$ is called complete if all Cauchy sequences converge (to $\hat{x}\in X$).
  • Examples:
    • Euclidean metric/norm: $X=\mathbb{R}^n, d(x,y) = ||x-y||_2$
    • $X = (1,3] \cup (4,\infty), d(x,y) = |x-y|$
      • $A = (1,3]$ is both open and closed, $\partial A = \{3\}$
    • $X=[0,3], d(x,y) = |x-y|$
      • complete
    • $X=(0,3), d(x,y) = \delta_{x,y}$
      • complete

Vector Spaces

Normed Spaces (distance + length)

• $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Let $X$ be a $\mathbb{F}$-vector space.
• Norm: A map $||\cdot||: X\rightarrow [0, \infty)$ is called norm if
  • $||x|| = 0 \Leftrightarrow x = 0$ (positive definite)
  • $||\lambda x|| = |\lambda|||x||, \forall \lambda\in\mathbb{F}, x\in X$ (absolutely homogeneous)
  • $||x+y||\le ||x|| + ||y||$ (triangle inequality)
  • (not necessarily defined by an inner product)
• Normed space: $(X, ||\cdot||)$
  • $d_{||\cdot||}(x,y):= ||x-y||$ defines a metric for the set $X$.
• Banach space: the normed space $(X, ||\cdot||)$ is called a Banach space if $(X, d_{||\cdot||})$ is a complete metric space.
• Examples:
  • $X = \mathbb{R}, d = |x-y|$ is a metric. Banach space.
  • $X =\{0\}, d = |0|=0$, zero-dimensioned real vector space. Banach space.
    • Theorem: A normed vector space $V$ is a Banach space iff every absolutely convergent series is convergent.
  • $\mathbb{R}^n, \mathbb{C}^n$ with respect to any of the $\lVert\cdot \rVert_p$ norms. Banach
  • The space of bounded continous functions $C_\infty(X)$. Banach.
  • $l^p$ space: $l^p(\mathbb{N}, \mathbb{F}), p\in[1,\infty)$ be defined as all sequences in $\mathbb{F}$ such that $\sum_{n=1}^\infty |x_n|^p< \infty$ (converges)
    • $\lVert\cdot\rVert_p: l^p\rightarrow [0,\infty)$ with $\lVert x\rVert_p:= \left(\sum_{n=1}^\infty |x_n|^p\right)^\frac{1}{p}$ is a norm.
    • $(l^p,\lVert\cdot\rVert_p)$ is a Banach space.

Inner Product Spaces (distance, length, angle)

• $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Let $X$ be a $\mathbb{F}$-vector space.
• Inner Product: $\langle\cdot,\cdot\rangle: X\times X\rightarrow \mathbb{F}$. $\langle u,v\rangle_{standard} = \sum_i \overline{u_i}v_i$
  • Positive definite: $\langle x,x\rangle\ge 0 (\in\mathbb{R})$ for all $x\in V$. $\langle x,x\rangle=0\Leftrightarrow x=\mathbf{0}$
  • Linear in the second argument:
    • $\langle x,y+z\rangle = \langle x,y\rangle +\langle x,z\rangle$
    • $\langle x,\lambda\cdot y\rangle = \lambda \langle x,y\rangle$
  • conjugate symmetric: $\langle x,y\rangle = \overline{\langle y, x\rangle}$
• Inner Product Space: $(X, \langle\cdot,\cdot\rangle)$
  • $\lVert x\rVert_{\langle , \rangle}:= \sqrt{\langle x, x \rangle}$ defines a norm
• Hilbert Space: the inner product space $(X, \langle\cdot,\cdot\rangle)$ is called a Hilber space if $(X, \lVert \cdot \rVert_{\langle , \rangle})$ is a Banach space (the associated metric space is complete). Examples:
  • $\mathbb{R}^n, \mathbb{C}^n$ with $\langle x,y\rangle = \sum_{i=1}^n \overline{x_i}y_i$ (Euclidean)
  • $L^2(\mathbb{N}, \mathbb{F})$ with $\langle x,y\rangle = \sum_{i=1}^\infty \overline{x_i}y_i$ (infinite dimension)
  • $C([0,1], \mathbb{F})$ with $\langle f,g\rangle = \int_0^1\overline{f(t)}g(t)dt$ (continuous function space, not complete, not Hilbert)
• Orthogonality: $x\perp y$ if $\langle x,y\rangle = 0$
  • Subsets $U,V\subseteq X$, $U\perp V$ if $x\perp y$ for all $x\in U$, $y\in V$.
  • Orthogonal complement: $U^\perp = \{x\in X| \langle x,u\rangle = 0, \forall u\in U\}$ for a subset $U\subseteq X$.
    • The complement is always a closed subspace.
    • $U\subseteq V \Rightarrow U^\perp \subseteq V^\perp$
    • $\{0\}^\perp = X, X^\perp = \{0\}$

Operators

• Continuity: $(X,d_X), (Y,d_Y)$ two metric spaces. A map $f:X\rightarrow Y$ is called continous if $f^{-1}[B]$ is open in $X$ for all open sets $B\in Y$.
  • Sequentially continous: if for all $x\in X$ and all sequences in X converges to $x$ holds $f(x_n)\rightarrow f(x)$
  • For metric spaces, continous and sequentially continuous are equivalent.
  • Examples:
    • $(X, d_X)$ discrete metric space, $(Y, d_Y)$ any metric space $\Rightarrow$ all $f:X\rightarrow Y$ are continous.
    • $f[X] = y_0\in Y$ is always continous
    • normed space $f(x) = \lVert x\rVert$ is continous
    • inner product space $f(x) = \langle x_0, x\rangle$ is continous
• Operator: $T:X\rightarrow Y$
  • linearity: conserves the algebraic structure
  • continuity: conserves the topological structure
  • Operator norm: two normed spaces, $T: X\rightarrow Y$ linear. $\lVert T\rVert:= \sup \{\frac{\lVert Tx\rVert_Y}{\lVert x\rVert_X}| x\in X, x\ne 0\}$ is called the operator norm. Bounded if $\lVert T\rVert<\infty$.
    • Equivalences: T is continous == continous at 0 == bounded.