Fundamental Theorems in Functional Analysis

Def: A seminorm or prenorm on a vector space $X$ is real valued function $p:X\to \mathbb{R}$ such that the following conditions are satisfies by all $x,y\in X$ and $\alpha \in \mathbb{F}$.

• $p(\alpha x)=|\alpha |p(x).$
• $p(x+y)\le p(x)+p(y)$.
  From this definition we get that $p(0)=0$ and $p(x)\ge 0$ for all $x\in X$.

Def: A function $f:X\to {\mathbb{R}}^{\ge 0}$ is countably subadditive if $f(\sum x_n)\le \sum f(x_n)$ for every convergent series $\sum x_n$ in $X$.

Zabreĭko's Lemma: Every countable subadditive seminorm on a Banach space is continuous.

Open Mapping Theorem: Every surjective bounded linear operator from a Banach space is an open mapping.

Cor: Every bijective bounded linear operator from a Banach space into a Banach space is an isomorphism.

Def: Two norms on the same vector space are equivalent if they induce the same topology.

Cor: Suppose $\parallel \cdot {\parallel }_1$ and $\parallel \cdot {\parallel }_2$ are two Banach norms on a vector space $X$ and that the identity from $(X,\parallel \cdot {\parallel }_1)$ to $(X,\parallel \cdot {\parallel }_2)$ is continuous, then the two norms are equivalent.

Def: A family of $\mathcal{F}$ of linear operator from a normed space $X$ into a normed space $Y$ is said to be pointwise bounded if, for each element of $x\in X$, the set $\{Tx\mid T\in \mathcal{F}\}$ is bounded, and is said to be uniformly bounded, if for each bounded set $B\subseteq X$, the set $\bigcup \{T[B]\mid T\in \mathcal{F}\}$ is bounded.

The Uniform Boundedness Principle: Let $\mathcal{F}\subseteq \mathcal{B}(X,Y)$, where $X$ is a Banach spaces and $Y$ a normed space. If $sup\{\parallel Tx\parallel \mid T\in \mathcal{F}\}$ is finite for each $x\in X$, then $sup\{\parallel T\parallel \mid T\in \mathcal{F}\}$ is finite.

Cor: Let $(T_n)$ be a sequence of bounded linear operators from a Banach space $X$ into a normed space $Y$ such that $lim{T}_{n}X$ exists for each $x\in X$. Define $T:X\to Y$ by the formula $Tx:=lim{T}_{n}x$. Then $T$ is a bounded linear operator from $X$ into $Y$.

The Closed Graph Theorem: Let $T$ be a linear operator from a Banach space $X$ into a Banach space $Y$. Suppose that whenever a sequence $(x_n{)}_{n<\omega }$ in $X$ converges to some $x\in X$ and $(T{x}_n{)}_{n<\omega }$ converges to some $y\in Y$, it follows that $Tx=y$. Then $T$ is bounded. Meaning that, if the set $\{(x,T(x))\mid x\in X\}\subseteq X\times Y$ is a closed subset, then $T$ is bounded.

Def: Let $A$ be an absorbing subset of a vector space $X$. For each $x\in X$, let $p_A(x):=inf\{t>0\mid t>0\wedge x\in tA\}$. Then $p_A$ is the Minkowski functional or gauge functional of $A$.

Prop: