Inner product - Revision history

Rongzhou at 20:41, 30 September 2026

2026-09-30T20:41:32Z

← Older revision		Revision as of 20:41, 30 September 2026
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	~~An inner product on~~ $V$ is a ~~map~~		== Mathematical definition ==

			Let $V$ be a vector space over $\mathbb{F}$.

			Map

	$$		$$
	\langle \cdot, \cdot \rangle : V \times V \to \mathbb{F}, \quad (u, v) \mapsto \langle u, v \rangle		\begin{aligned}
			\langle \cdot, \cdot \rangle : V \times V \to \mathbb{F}\\
			\quad (u, v) \mapsto \langle u, v \rangle
			\end{aligned}
	$$		$$

	~~It must satisfies four properties~~:		is an '''inner product''' on $V$ if 4 conditions are satisfied:

	* Linearity in the 1st slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ ~~for all~~ $u, v~~, w~~ \~~in V; a~~ \in \~~mathbb{F}~~$.		* '''Linearity in the 1st slot''': $\forall (u, v, w \in V, a \in \mathbb{F}), \langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$
	* Positivity: $\~~langle~~ v, v \~~rangle~~ \~~geq~~ 0~~$ for all $v \in V$.~~		* '''Definiteness''': $\langle v, v \rangle = 0 \iff v = \vec 0$.
	* Definiteness: $\langle v, v \rangle = 0$ ~~if and only if $v = 0$.~~		* '''Positivity''': $\forall v \in V, v \neq \vec 0, \langle v, v \rangle > 0$
	* Conjugate symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}~~$ for all $u, v \in V~~$.		* '''Conjugate symmetry''': $\forall u, v \in V, \langle u, v \rangle = \overline{\langle v, u \rangle}$.

	<blockquote>		<blockquote>
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	<blockquote>		<blockquote>
	~~Note~~: confusingly, the convention in physics is often the exact opposite: an inner product is linear in the second slot and anti-linear in the first slot.		'''Warning''': confusingly, the convention in physics is often the exact opposite: an inner product is linear in the second slot and anti-linear in the first slot.
	</blockquote>		</blockquote>

Rongzhou at 20:30, 30 September 2026

2026-09-30T20:30:49Z

← Older revision		Revision as of 20:30, 30 September 2026
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	* Linearity in the 1st slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ for all $u, v, w \in V; a \in \mathbb{F}$.		* Linearity in the 1st slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ for all $u, v, w \in V; a \in \mathbb{F}$.
	* Positivity: $\langle v, v \rangle \geq 0$ for all $v \in V$.		* Positivity: $\langle v, v \rangle \geq 0$ for all $v \in V$.
	* ~~Positive definiteness~~: $\langle v, v \rangle = 0$ if and only if $v = 0$.		* Definiteness: $\langle v, v \rangle = 0$ if and only if $v = 0$.
	* Conjugate symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all $u, v \in V$.		* Conjugate symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all $u, v \in V$.

Rongzhou: physics convention opposite

2026-09-30T18:44:24Z

physics convention opposite

← Older revision		Revision as of 18:44, 30 September 2026
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	Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.		Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
			</blockquote>

			<blockquote>
			Note: confusingly, the convention in physics is often the exact opposite: an inner product is linear in the second slot and anti-linear in the first slot.
	</blockquote>		</blockquote>

Rongzhou at 08:48, 30 September 2026

2026-09-30T08:48:50Z

@@ Line 19: / Line 19: @@
 Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
 </blockquote>
 == See also ==
 [[Inner product space]]

Rongzhou at 08:46, 30 September 2026

2026-09-30T08:46:46Z

← Older revision		Revision as of 08:46, 30 September 2026
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	<blockquote>		<blockquote>
	Recall $$\forall x\in\mathbb{R},\; x=\overline{x}$$. Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.		Recall:

			$$\forall x\in\mathbb{R},\; x=\overline{x}$$

			Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
	</blockquote>		</blockquote>

Rongzhou at 14:20, 29 September 2026

2026-09-29T14:20:19Z

← Older revision		Revision as of 14:20, 29 September 2026
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	It must satisfies four properties:		It must satisfies four properties:

	* Linearity in ~~first~~ slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ for all $u, v, w \in V; a \in \mathbb{F}$.		* Linearity in the 1st slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ for all $u, v, w \in V; a \in \mathbb{F}$.
	* Positivity: $\langle v, v \rangle \geq 0$ for all $v \in V$.		* Positivity: $\langle v, v \rangle \geq 0$ for all $v \in V$.
	* Positive definiteness: $\langle v, v \rangle = 0$ if and only if $v = 0$.		* Positive definiteness: $\langle v, v \rangle = 0$ if and only if $v = 0$.

Rongzhou: /* See also */

2026-09-29T13:05:28Z

Rongzhou at 12:49, 29 September 2026

2026-09-29T12:49:53Z

← Older revision		Revision as of 12:49, 29 September 2026
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	<blockquote>		<blockquote>
	Recall ~~that every real number~~ $x \in \mathbb{R}$ ~~equals its complex conjugate~~. Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.		Recall $$\forall x\in\mathbb{R},\; x=\overline{x}$$. Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
	</blockquote>		</blockquote>

Rongzhou at 12:35, 29 September 2026

2026-09-29T12:35:45Z

← Older revision	Revision as of 12:35, 29 September 2026
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		An inner product on $V$ is a map

		$$
		\langle \cdot, \cdot \rangle : V \times V \to \mathbb{F}, \quad (u, v) \mapsto \langle u, v \rangle
		$$

		It must satisfies four properties:

		* Linearity in first slot: $\langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ and $\langle au, v \rangle = a \langle u, v \rangle$ for all $u, v, w \in V; a \in \mathbb{F}$.
		* Positivity: $\langle v, v \rangle \geq 0$ for all $v \in V$.
		* Positive definiteness: $\langle v, v \rangle = 0$ if and only if $v = 0$.
		* Conjugate symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all $u, v \in V$.

		<blockquote>
		Recall that every real number $x \in \mathbb{R}$ equals its complex conjugate. Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.
		</blockquote>

		== See also ==

		[[Inner vector space]]

Rongzhou: Created the wiki page for inner product

2026-09-29T11:02:39Z

Created the wiki page for inner product

New page

← Older revision		Revision as of 13:05, 29 September 2026
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	== See also ==		== See also ==

	[[Inner ~~vector~~ space]]		[[Inner product space]]