Notes on Normed Linear & Banach Space
Topological vector space $(X, (+, \cdot_\mathbb{F}), \mathcal{T})$ is a vector space over a topological field $(\mathbb{F}, \mathcal{T}_F)$ equipped with a topology such that vector addition and scalar multiplication are continuous maps.
Continuous operator $A: M \to Y$ between (subsets of) topological vector spaces is a continuous map: $A \in C(M, Y)$. Weakly continuous operator... Strongly continuous operator... Continuous functional $f: X \to \mathbb{F}$ on a topological vector space is a continuous operator to its underlying scalar field: $f \in C(X)$. Dual space or adjoint space $X^∗$ of a topological vector space is the vector space of continuous linear functionals: $X^∗ = \mathcal{L}(X) \cap C(X)$. Similar to vector space, we will call any element in the dual space of a topological vector space a covector, i.e. a continuous linear functional.
Locally convex topology on a vector space is one that has a local base at the origin consisting of balanced, convex, and absorbing sets. Locally convex space is a topological vector space whose topology is Hausdorff locally convex. The dual space of a locally convex space separate the points of the space.(?) 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 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 if and only if the subset is a total set, i.e. the subset separates the points of the space. 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.
Scalar product $q: V \times W \to \mathbb{F}$ on the Cartesian product of two vector spaces over the same field is a nondegenerate bilinear scalar-valued function: $q(a v + b v', w) = a q(v, w) + b q(v', w)$, $q(v, \cdot) = 0 \implies v = 0$, and both apply analogously for the second variable. Dual pair of vector spaces $(V, W, q(\cdot, \cdot))$ is a pair of vector spaces $(V, W)$ over the same field $\mathbb{F}$, together with a scalar product $q(\cdot, \cdot)$. Weak topology defined by a dual pair of vector spaces over the same topological field is the weakest topology on one space such that the scalar product with any vector in the other space is a continuous linear functional: $\{q(\cdot, w) : w \in W\} \subset V^∗$. Strong topology defined by a dual pair of vector spaces over the same topological field is the topology on one space of uniform convergence on the bounded subsets of the other space 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. Second dual space or bi-dual 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 $X$, i.e. a locally convex space with several properties of Banach spaces and Fréchet spaces, is an invertible linear operator that maps each point to the evaluation map of continuous linear functionals at that point: $(\pi v)(\omega) = \omega(v)$. Semi-reflexive space is a Hausdorff, locally convex space that coincides with its second dual: $X = X^{∗∗}$.
Adjoint operator, dual operator, or conjugate operator $A^∗: Y^∗ \to X^∗$ of a linear operator $A: X \to Y$ between locally convex spaces is a linear operator between the strong dual spaces $Y^∗$ and $X^∗$ such that $(y^∗, x^∗) \in A^∗$ if and only if $\exists (x, y) \in A$: $(y, y^∗) = (x, x^∗)$. The adjoint operator of a continuous linear operator is also continuous.
Total subset or fundamental subset in a topological vector space is a subset whose span is a dense subspace: $\overline{\text{Span}(A)} = X$.
Compact operator between topological vector spaces is an operator that maps bounded subsets to totally-bounded/pre-compact subsets.
Norm $\|\cdot\|: X \to \mathbb{R}_{\ge 0}$ on a real or complex vector space $X$ is a nonnegative operator that satisfies: (1) non-degeneracy: nonzero vector has nonzero norm; $\|x\| = 0 \implies x = 0$; (2) homogeneity: scaled vector has absolutely scaled norm; $\|a x\| = |a| \|x\|$; (3) triangle axiom/inequality: norm of sum does not exceed sum of norms; $\|x + y\| \le \|x\| + \|y\|$.
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.
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.
Subspace $(S, (+, \cdot_\mathbb{F}), \|\cdot\|)$ of a normed space is a normed space consisting of a linear subspace and the norm restricted to the subspace. A subspace of a Banach space is a Banach space if and only if it is a closed subset. Every finite dimensional subspace of a normed space is a complete metric space. In particular, every finite dimensional normed space is a complete metric space.
The canonical embedding $\pi: X \to X^{∗∗}$ of a normed space is a linear isometry. Reflexive space is a Banach space whose canonical embedding coincides with its second dual space: $\pi(X) = X^{∗∗}$.
Operator norm of an operator between two normed spaces is the supremum of norms of the image of unit ball: $\|A\| = \sup_{\|x\| \le 1} \|Ax\|$. Bounded linear operator between normed spaces is a linear operator with a finite operator norm: $\|A\| < \infty$. Space of bounded linear operators $B(X, Y)$ between two normed spaces is the normed space consisting of the set of all bounded linear operators between the given spaces and the operator norm. A bounded linear operator space is Banach if the codomain is Banach. A linear operator between Banach spaces is continuous if and only if it is bounded: if $\hat X = X$, $\hat Y = Y$, $A \in \mathcal{L}(X, Y)$, then $A \in C(X, Y)$ iff $A \in B(X, Y)$. In particular, the set of all bounded linear functionals on a Banach space equals its dual space: $\hat X = X$ then $X^∗ = \mathcal{B}(X)$.
Completely-continuous operator from a Banach space to a normed space is a continuous linear operator that maps weakly-convergent sequences to norm-convergent sequences. Compact operators are completely-continuous; completely-continuous operators are compact if the codomain is also Banach. The adjoint operator of a completely-continuous operator is also completely-continuous.
Fundamental theorems/principles of linear operators on normed and Banach spaces.