4.1 Seminorm

Definition 4.1.1 (Topological Vector Space).label Let $E$ be a vector space over $K\in \curl{\R,\com}$ and $\cali{T}\suf 2^{E}$ be a topology. We say that the pair $(E,\cali{T})$ is a topological vector space if the following holds:

1. (TV1)

   $E\ti E\to E$ defined by $(x,y)\mto x+y$ is continuous

2. (TV2)

   $K\ti E\to E$ defined by $(\lam,x)\mto \lam x$ is continuous

Definition 4.1.2 (Convex).label Let $E$ be a vector space over $K\in \curl{\R,\com}$ then $A\suf E$ is convex if for any $x,y\in A$, $\curl{\lam x+(1-\lam)y:\lam\in [0,1]}\suf A$

Definition 4.1.3 (Sublinear Functional).label Let $E$ be a vector space over $K\in \curl{\R,\com}$, then a sublinear functional is a mapping $\rho:E\to\R$ such that:

1. (1)

   $\rho(0)=0$

2. (2)

   $\fall x\in E$ and $\lam\geq 0,\rho(\lam x)=\lam\rho(x)$

3. (3)

   $\fall x,y\in E,\rho(x+y)\leq \rho(x)+\rho(y)$

Definition 4.1.4 (Seminorm).label Let $E$ be a vector space over $K\in \curl{\R,\com}$, then a seminorm on $E$ is a mapping $\rho:E\to[0,\infty)$ such that:

1. (SN1)

   $\rho(0)=0$

2. (SN2)

   $\fall x\in E,\fall \lam\in K,\rho(\lam x)=\abs{\lam}\rho(x)$

3. (SN3)

   $\fall x,y\in E,\rho(x+y)\leq \rho(x)+\rho(y)$

Lemma 4.1.5 (Seminorm Properties).label Let $E$ be a TVS over $K\in \curl{\R,\com}$ and $\rho:E\to[0,\infty)$ be a seminorm on $E$, then the following are equivalent:

1. (1)

   $\rho$ is uniformly continuous

2. (2)

   $\rho$ is continuous

3. (3)

   $\rho$ is continuous at $0$

4. (4)

   $\curl{x\in E:\rho(x)<1}\in \cali{N}_{E}(0)$

Definition 4.1.6 (Topology Induced by Seminorm).label Let $E$ be a vector space over $K\in \curl{\R,\com}$ and $\curl{\rho_i}_{i\in I}$ be a collection of seminorms then:

1. (1)

   For each $i\in I$, $d_{i}:E\ti E\to [0,\infty)$ defined by $(x,y)\mto \rho_{i}(x-y)$ is a pseudo-metric

2. (2)

   The topology induced by $\curl{d_i}_{i\in I}$ makes $E$ a topological vector space on which $\rho_{i}:E\to[0,\infty)$ is continuous for each $i\in I$

1. (1)

   For any family of $\curl{\rho_j}_{j\in J}$ of continuous seminorms on $E$, the vector space topology induced by $\curl{\rho_j}_{j\in J}$ is contained in the vector space topology induced by $\curl{\rho_i}_{i\in I}$

Definition 4.1.7 (Locally Convex Space).label Let $E$ be a TVS over $\curl{\R,\com}$ then $E$ is a locally convex space if there exists a family of seminorms $\curl{\rho_i}_{i\in I}$ that induces the topology on $E$