Notes on Normed Linear & Banach Space

Normed space and Banach space.

Topological vector space $(X, (+, \cdot_\mathbb{F}), \tau)$ is a vector space $(X, (+, \cdot))$ over a topological field $F$ equipped with a topology $\tau$ that is compatible with the vector space structure: vector addition $+$ and scalar multiplication $\cdot$ are continuous.

## Norm

A norm is a nonnegative homogeneous mapping on a vector space that satisfies the triangle axiom: for a vector space $X$ with underlying field $\mathbb{F}$, a mapping $\|\|: X \to \mathbb{R}_{\ge 0}$

• Non-degeneracy: $\|x\| = 0 \Leftrightarrow x = 0$;
• Homogeneity: $\|a x\| = |a| \|x\|, \forall a \in \mathbb{F}$;
• Triangle axiom: $\|x + y\| \le \|x\| + \|y\|$;

A 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, $d(x,y) = \|x − y\|$.

A Banach space is a complete normed space (in the induced metric).

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.

## Linear Operator

Bounded linear operator (the image of any bounded set in $X$ is bounded in $Y$). Compact linear operator (the image of any bounded set in $X$ is pre-compact in $Y$).

Continuous operator (topological vector spaces). Weakly continuous operator. Strongly continuous operator. Compact operator, aka completely-continuous operator.

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

All continuous linear operators $B(X, Y)$ between two given normed spaces $X$ and $Y$, together with the operator norm, is a normed space. The normed space $B(X, Y)$ is Banach if $Y$ is Banach.

A linear operator between two Banach spaces is continuous iff it is bounded, i.e. its operator norm is finite.

Three fundamental principles of linear analysis:

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;