inner product

[[concept]]

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Inner Product

An inner product on a vector space $V$ over field $K$ is a function $⟨, ⟩ : V \times V \to K$ such that for all $x, y, z \in V$ and all $c \in K$ we have

$⟨ x, x ⟩$ is real, nonnegative. Also, $⟨ x, x ⟩ = 0 ⟺ x = 0$
$⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩$
$⟨ c x, z ⟩ = c ⟨ x, z ⟩$ (2 and 3 indicate that this is linear)
$⟨ x, z ⟩ = \bar{⟨ z, x ⟩}$ (ie symmetric)

If all of these hold, then $V$ is an inner product space.

Example

$C^{n}$ over $C$ .

Euclidean inner product: $⟨ x, z ⟩ := z^{*} x$
some $A \in M_{n}$ positive definite: $⟨ x, z ⟩_{A} := z^{*} A x$
Note that the euclidean inner product is a special case of the second, where $A$ is the identity!

Mentions

const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
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	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

Created 2025-09-04 ֍ Last Modified 2025-11-14