# inner product

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## 1—10 of 11 matching pages

##### 1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
A complex linear vector space $V$ is called an inner product space if an inner product $\left\langle u,v\right\rangle\in\mathbb{C}$ is defined for all $u,v\in V$ with the properties: (i) $\left\langle u,v\right\rangle$ is complex linear in $u$; (ii) $\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$; (iii) $\left\langle v,v\right\rangle\geq 0$; (iv) if $\left\langle v,v\right\rangle=0$ then $v=0$. …Two elements $u$ and $v$ in $V$ are orthogonal if $\left\langle u,v\right\rangle=0$. …
1.18.3 $c_{n}=\left\langle v,v_{n}\right\rangle.$
1.18.12 $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)\overline{g(x)}\,\mathrm{d}% \alpha(x),$
The adjoint ${T}^{*}$ of $T$ does satisfy $\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle$ where $\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x$. …
##### 2: 1.2 Elementary Algebra
1.2.41 $\left\langle\mathbf{u},\mathbf{v}\right\rangle=\overline{\left\langle\mathbf{v% },\mathbf{u}\right\rangle},$
1.2.42 $\left\langle\alpha\mathbf{u},\beta\mathbf{v}\right\rangle=\alpha\overline{% \beta}\left\langle\mathbf{u},\mathbf{v}\right\rangle,$
1.2.43 $\left\langle\mathbf{v},\mathbf{v}\right\rangle=0,$
1.2.46 $\left\|{\mathbf{v}}\right\|=\left\|{\mathbf{v}}\right\|_{2}=\sqrt{\left\langle% \mathbf{v},\mathbf{v}\right\rangle},$
##### 3: 1.1 Special Notation
 $x,y$ real variables. … inner, or scalar, product for real or complex vectors or functions. …
##### 4: 1.3 Determinants, Linear Operators, and Spectral Expansions
The adjoint of a matrix $\mathbf{A}$ is the matrix ${\mathbf{A}}^{*}$ such that $\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle$ for all $\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}$. …
1.3.20 $\mathbf{u}=\sum_{i=1}^{n}c_{i}\mathbf{a}_{i},$ $c_{i}=\left\langle\mathbf{u},\mathbf{a}_{i}\right\rangle$.
##### 5: 18.39 Applications in the Physical Sciences
$c_{n}=\left\langle\chi,\psi_{n}\right\rangle,$
##### 6: 29.14 Orthogonality
is orthogonal and complete with respect to the inner productEach of the following seven systems is orthogonal and complete with respect to the inner product (29.14.2): …When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product
##### 7: Bibliography I
• A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.
• ##### 8: 18.36 Miscellaneous Polynomials
Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. …
##### 9: 31.15 Stieltjes Polynomials
with respect to the inner product
##### 10: 26.10 Integer Partitions: Other Restrictions
26.10.2 $\sum_{n=0}^{\infty}p\left(\mathcal{D},n\right)q^{n}=\prod_{j=1}^{\infty}(1+q^{% j})=\prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}=1+\sum_{m=1}^{\infty}\frac{q^{m(m% +1)/2}}{(1-q)(1-q^{2})\cdots(1-q^{m})}=1+\sum_{m=1}^{\infty}q^{m}(1+q)(1+q^{2}% )\cdots\*(1+q^{m-1}),$
26.10.3 $(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,\mathcal{D},n\right)x^{m}q^{n}=% \sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{% k}(1+x\,q^{j}),$ $|x|<1$,
26.10.5 $\sum_{n=0}^{\infty}p\left(\in\!S,n\right)q^{n}=\prod_{j\in S}\frac{1}{1-q^{j}}.$
where the inner sum is the sum of all positive odd divisors of $t$. … where the inner sum is the sum of all positive divisors of $t$ that are in $S$. …