512: Norm Induced by Inner Product on Real or Complex Vectors Space

<The previous article in this series | The table of contents of this series | The next article in this series>

definition of norm induced by inner product on real or complex vectors space

---

Topics

About: normed vectors space

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

---

Starting Context

• The reader knows a definition of inner product on real or complex vectors space.
• The reader knows a definition of norm on real or complex vectors space.

---

Target Context

• The reader will have a definition of norm induced by inner product on real or complex vectors space.

---

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

---

Main Body

---

1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\mathbb{R},\mathbb{C}\}$, with the canonical field structure
$V$: $\in \{\text{ the vectors fields over }F\}$ with any inner product, $⟨\bullet ,\bullet ⟩$
$\ast \Vert \bullet \Vert$: $:V\to \mathbb{R},v\mapsto \sqrt{⟨v,v⟩}$
//

Conditions:
//

---

2: Natural Language Description

For the field, $F\in \{\mathbb{R},\mathbb{C}\}$, any vectors space, $V$, over $F$, with any inner product, $⟨\bullet ,\bullet ⟩$, the norm, $\Vert \bullet \Vert :V\to \mathbb{R},v\mapsto \sqrt{⟨v,v⟩}$

---

3: Note

It is indeed a norm, by the proposition that any inner product on any real or complex vectors space induces a norm.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>