Quasinorm

Definition:[1] A quasinorm on a vector space ${\displaystyle X}$ is a real-valued map ${\displaystyle p}$ on ${\displaystyle X}$ that satisfies the following conditions:

1. Non-negativity: ${\displaystyle p\geq 0;}$
2. Absolute homogeneity: ${\displaystyle p(sx)=|s|p(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s;}$
3. there exists a ${\displaystyle k\geq 1}$ such that ${\displaystyle p(x+y)\leq k[p(x)+p(y)]}$ for all ${\displaystyle x,y\in X.}$

If ${\displaystyle p}$ is a quasinorm on ${\displaystyle X}$ then ${\displaystyle p}$ induces a vector topology on ${\displaystyle X}$ whose neighborhood basis at the origin is given by the sets:[1]

$${\displaystyle \{x\in X:p(x)<1/n\}}$$

as ${\displaystyle n}$ ranges over the positive integers. A topological vector space (TVS) with such a topology is called a quasinormed space.

Every quasinormed TVS is a pseudometrizable.

A vector space with an associated quasinorm is called a quasinormed vector space.

A complete quasinormed space is called a quasi-Banach space.

A quasinormed space ${\displaystyle (A,\|\,\cdot \,\|)}$ is called a quasinormed algebra if the vector space ${\displaystyle A}$ is an algebra and there is a constant ${\displaystyle K>0}$ such that

$${\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}$$

for all ${\displaystyle x,y\in A.}$

A complete quasinormed algebra is called a quasi-Banach algebra.

A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.[1]

• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Seminorm
• Topological vector space – Vector space with a notion of nearness

• Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology. Springer. ISBN 0-7923-6970-X.
• Conway, John B. (1990). A Course in Functional Analysis. Springer. ISBN 0-387-97245-5.
• Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. Vol. 19. Springer. ISBN 3-540-50584-9.
• Swartz, Charles (1992). An Introduction to Functional Analysis. CRC Press. ISBN 0-8247-8643-2.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.