Notes on Normed Linear & Banach Space

## Topological Vector Space

Topological vector space $(X, (+, \cdot_\mathbb{F}), \mathcal{T})$ is a vector space $(X, (+, \cdot))$ over a topological field $F$ equipped with a topology $\mathcal{T}$ that is compatible with its algebraic structure, i.e. vector addition $+$ and scalar multiplication $\cdot$ are continuous. The topology in a topological vector space is locally convex if it has a local base at the origin $0$ consisting of balanced, convex, and absorbing sets. Locally convex space is a topological vector space whose topology is Hausdorff locally convex.

Dual space (or adjoint space) $X^∗$ of a topological vector space $X$ is the vector space of continuous linear functionals $\alpha: X \to \mathbb{F}$ from $X$ to its underlying field $\mathbb{F}$. If $X$ is a locally convex space, then the functionals $\alpha \in X^∗$ separate the points of $X$.

Semi-norm $p: X \to \mathbb{R}_{\ge 0}$ is a mapping that meets the requirements of a norm except for non-degeneracy: (1) homogeneity: $p(a x) = |a| p(x)$; (2) triangle axiom: $p(x + y) \le p(x) + p(y)$. Weak topology $\sigma(X, F)$ on a topological vector space $X$ given a subset $F$ of its adjoint space $X^∗$ is the locally convex topology generated by the family of semi-norms $\{|f(\cdot)|\}_{f \in F}$; it is a Hausdorff topology iff $F$ is a total set, i.e. $F$ separates the points of $X$. Weak∗ topology on the adjoint space $X^∗$ of a topological vector space $X$ is the weakest topology on $X^∗$ for which all the evaluation mappings $A_x: X^∗ \to \mathbb{F}$, $A_x f = f(x)$, $x \in X$ are continuous.

Dual pair of vector spaces $(L, M, \phi(\cdot, \cdot))$ is a pair of vector spaces $L$ and $M$ over a field $\mathbb{F}$, together with a non-degenerate bilinear form $\phi: L \times M \to \mathbb{F}$: (1) bilinearity: $\phi(a l + b l', m) = a \phi(l, m) + b \phi(l', m)$ and $\phi(l, a m + b m') = a \phi(l, m) + b \phi(l, m')$; (2) non-degeneracy: $\phi(l, \cdot) = 0 \Rightarrow l = 0$ and $\phi(\cdot, m) = 0 \Rightarrow m = 0$. Weak topology defined by a dual pair of vector spaces $(L, M, \phi(\cdot, \cdot))$, given a topology $\mathcal{T}$ on the underlying field $\mathbb{F}$, is the weakest topology on $L$ (and analogously on $M$) such that the family of mappings $\{\phi(\cdot, m)\}_{m \in M}$ are continuous. Strong topology defined by a dual pair of vector spaces $(L, M, \phi(\cdot, \cdot))$ is the topology on $M$ (and analogously on $L$) of uniform convergence on the bounded subsets of $L$ for the weak topology defined by the dual pair.

Strong dual space $(X^∗, \mathcal{T})$ of a topological vector space $X$ is its dual space $X^∗$ with the strong topology $\mathcal{T}$. Second dual space (or bidual space) $X^{∗∗}$ of a Hausdorff locally convex space $X$ is the dual space of its strong dual space. Canonical embedding $\pi: X \to X^{∗∗}$ of a barrelled space $X$ (a locally convex space with several properties of Banach spaces and Fréchet spaces) is an invertible linear operator such that $\pi x: X^∗ \to \mathbb{F}$ is the evaluation mapping of $f \in X^∗$ at $x$.

Semi-reflexive space is a Hausdorff locally convex space $X$ that coincides with its second dual space $X^{∗∗}$.

Compact operator is an operator between two topological vector spaces $X$ and $Y$ such that the image of any bounded set in $X$ is totally-bounded/pre-compact in $Y$.

Continuous operator $A: X \to Y$ is a continuous mapping between two topological vector spaces; more generally, if the operator $A$ is only defined on a subset $M$ of $X$, it is continuous if it can be extended to a continuous operator from $X$: $\forall G \in \mathcal{T}_Y, \exists H \in \mathcal{T}_X: A^{-1}(G) = H \cap M$. Weakly continuous operator. Strongly continuous operator.

Adjoint operator (or dual operator, conjugate operator) $A^∗: Y^∗ \to X^∗$ of a linear operator $A: X \to Y$ between two locally convex spaces $X$ and $Y$ is a linear operator between the strong dual spaces $Y^∗$ and $X^∗$, such that: $(y^∗, x^∗) \in A^∗$, iff $\exists (x, y) \in A$: $(y, y^∗) = (x, x^∗)$. If $A$ is a continuous linear operator with domain $X$, then its adjoint operator $A^∗$ is also continuous.

## Norm

Norm $\|\cdot\|: X \to \mathbb{R}_{\ge 0}$ is a mapping from a vector space $X$ over the field $\mathbb{R}$ or $\mathbb{C}$ of real or complex numbers to nonnegative numbers $\mathbb{R}_{\ge 0}$, which satisfies:

1. Non-degeneracy: $\|x\| = 0 \Leftrightarrow x = 0$;
2. Homogeneity: $\|a x\| = |a| \|x\|$;
3. Triangle axiom: $\|x + y\| \le \|x\| + \|y\|$;

Normed space $(X, (+, \cdot_\mathbb{F}), \|\cdot\|)$ is a vector space with a norm. Norm specifies the length of each element of a vector space. A norm induces a metric on the vector space, $d(x,y) = \|x − y\|$, which in turn induces a topology on the vector space; as a result, normed spaces are topological vector spaces. For example, $L^p$ spaces are normed spaces.

Operator norm of an operator $A$ between two normed spaces $X$ and $Y$ is the supremum of norms of the image of unit ball: $\|A\| = \sup_{\|x\| \le 1} \|Ax\|$.

Banach space is a complete normed space (in the induced metric). For example, Sobolev space $W^{s,p}(\Omega)$ with norm $\|f\|_{s,p,\Omega} = \sum_{|\alpha| \le s} \|\partial_x^\alpha f\|_{L^p(\Omega)}$ is a Banach space. The dual space $X^∗$ of a normed space $X$, together with norm $\|f\| = \sup_{x \ne 0} \frac{|f(x)|}{\|x\|}$, is a Banach space.

Reflexive space is a Banach space $X$ whose canonical embedding $\pi(X)$ coincides with its second dual space $X^{∗∗}$.

## Linear Operator

Linear operators between infinite-dimensional vector spaces are studied under the assumption that they are continuous with respect to certain topologies. All continuous linear operators $\mathcal{B}(X, Y)$ between two given normed spaces $X$ and $Y$, together with the operator norm, is a normed space; moreover, $\mathcal{B}(X, Y)$ is Banach if $Y$ is Banach. A continuous linear operator $A$ between two normed spaces $X$ and $Y$ with domain $X$ has the same operator norm as its adjoint operator: $\|A^∗\| = \|A\|$. The canonical embedding $\pi: X \to X^{∗∗}$ of a normed space $X$ is an isometric linear operator.

Bounded linear operator is a linear operator $A: X \to Y$ between two normed spaces $X$ and $Y$ such that the image of any bounded set in $X$ is also bounded in $Y$, or equivalently, its operator norm $\|A\|$ is finite. A linear operator between two Banach spaces is continuous iff it is bounded.

Completely-continuous operator is a bounded linear operator $A: X \to Y$ where $X$ is Banach, such that weakly-convergent sequences in $X$ are mapped to norm-convergent sequences in $Y$. Compact operators are completely-continuous; if $Y$ is also Banach, completely-continuous operators are compact. The adjoint operator $A^∗$ of a completely-continuous operator $A$ is also completely-continuous.

Three fundamental principles of linear analysis (on Banach spaces):

1. "Uniform boundedness principle": (Banach–Steinhaus theorem) If a sequence of continuous linear operators between two given Banach spaces is point-wise bounded, their operator norms are also bounded; given $A_n: X \to Y, n \in \mathbb{N}$, if $\sup_n \|A_n x\| < \infty, \forall x \in X$, then $\sup_n \|A_n\| < \infty$;
2. "Open mapping principle": (Banach's theorem) If a continuous linear operator has an inverse, this inverse operator is automatically continuous;
3. (Hahn–Banach theorem);