Inner product space: Difference between revisions

Latest revision as of 09:18, 1 October 2026

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An inner product space is a vector space over $\mathbb{F}$ together with an inner product $\langle \cdot, \cdot \rangle$.

Let $V = \mathbb{F}^n$, $u = (u_1, \ldots, u_n), v = (v_1, \ldots, v_n) \in \mathbb{F}^n$. Then $V$ with

$$ \langle u, v \rangle = \sum_{i=1}^n u_i \overline{v}_i. $$

For $\mathbb{F} = \mathbb{R}$, $\overline{v}_i=v_i$

$$ u \cdot v = u_1 v_1 + \cdots + u_n v_n. $$

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-== Overview ==
+An inner product space is a [[vector space]] over $\mathbb{F}$ together with an [[inner product]] $\langle \cdot, \cdot \rangle$.
-mathematical space
+== Example ==
+Let $V = \mathbb{F}^n$, $u = (u_1, \ldots, u_n), v = (v_1, \ldots, v_n) \in \mathbb{F}^n$. Then $V$ with
+$$
+\langle u, v \rangle = \sum_{i=1}^n u_i \overline{v}_i.
+$$
+For $\mathbb{F} = \mathbb{R}$, $\overline{v}_i=v_i$
+$$
+u \cdot v = u_1 v_1 + \cdots + u_n v_n.
+$$
+== Properties ==
+* [[Cauchy-Schwarz inequality]]
+* [[Triangle inequality#Generalisation to vector spaces|Triangle inequality]]
+* [[Parallelogram law#Generalisation into vector spaces|Parallelogram law]]