# l2 space

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## 1—10 of 51 matching pages

##### 1: 1.2 Elementary Algebra
1.2.46 $\left\|{\mathbf{v}}\right\|=\left\|{\mathbf{v}}\right\|_{2}=\sqrt{\left\langle% \mathbf{v},\mathbf{v}\right\rangle},$
Let $\left\|{\mathbf{x}}\right\|=\left\|{\mathbf{x}}\right\|_{2}$ the $l^{2}$ norm, and $\mathbf{E}_{n}$ the space of all $n$-dimensional vectors. …
1.2.67 $\left\|{\mathbf{A}}\right\|=\max_{\mathbf{x}\in\mathbf{E}_{n}\setminus\left\{% \boldsymbol{{0}}\right\}}\frac{\left\|{\mathbf{A}\mathbf{x}}\right\|}{\left\|{% \mathbf{x}}\right\|}=\max_{\left\|{\mathbf{x}}\right\|=1}\left\|{\mathbf{A}% \mathbf{x}}\right\|.$
##### 2: 1.1 Special Notation
 $x,y$ real variables. … the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$. … the space of all $n$-dimensional vectors. …
##### 3: 20.12 Mathematical Applications
The space of complex tori $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_{1}$ and $z_{2}$ are regarded as equivalent if there exist integers $m,n$ such that $z_{1}-z_{2}=m+\tau n$) is mapped into the projective space $P^{3}$ via the identification $z\to(\theta_{1}\left(2z\middle|\tau\right),\theta_{2}\left(2z\middle|\tau% \right),\theta_{3}\left(2z\middle|\tau\right),\theta_{4}\left(2z\middle|\tau% \right))$. …This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …
##### 4: 25.17 Physical Applications
Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). …
##### 5: Brian D. Sleeman
He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. …
##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### §1.18(i) Hilbert spaces
With norm defined by …$V$ becomes a normed linear vector space. … …
 $a,b$ complex variables. … space of all real symmetric matrices. … space of positive-definite real symmetric matrices. … space of orthogonal matrices. …
35.5.2 $A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
35.5.4 $\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{\nu}A_{\nu}\left(\mathbf{S}\mathbf{X}\right)\,\mathrm% {d}{\mathbf{X}}=\operatorname{etr}\left(-\mathbf{S}\mathbf{T}^{-1}\right)\left% |\mathbf{T}\right|^{-\nu-\frac{1}{2}(m+1)},$ $\mathbf{S}\in\boldsymbol{\mathcal{S}}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$; $\Re\left(\nu\right)>-1$.
35.5.5 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left|\mathbf{T}-\mathbf{X}\right|% ^{\nu_{2}}\,\mathrm{d}{\mathbf{X}}=\left|\mathbf{T}\right|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),$ $\nu_{j}\in\mathbb{C}$, $\Re\left(\nu_{j}\right)>-1$, $j=1,2$; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$; $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.6 $B_{\nu}\left(\mathbf{T}\right)=\left|\mathbf{T}\right|^{-\nu}B_{-\nu}\left(% \mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.8 $\int_{\mathbf{O}(m)}\operatorname{etr}\left(\mathbf{S}\mathbf{H}\right)\mathrm% {d}{\mathbf{H}}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{% T}}\right)}{A_{-1/2}\left(\boldsymbol{{0}}\right)},$ $\mathbf{S}$ arbitrary.
Close to the origin $\mathbf{x}=\boldsymbol{{0}}$ of parameter space, the series in §36.8 can be used. …