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Linear Functionals Part II

  This is a continuation of the previous post:  Linear Functionals Part I Let us take a look at a Lemma... LEMMA:  Let $A=[a_{ij}]\in\mathbb{F}^{m\times n}$, that is either the real or complex scalar fields. Then the adjoint $A^*$ satisfies... $$\langle Ax,y\rangle=\langle x,A^*y\rangle\ \ \forall \ \ x\in\mathbb{F}^n,y\in\mathbb{F}^m$$ PROOF: Let us write this out... $$\begin{align*} Ax=\begin{pmatrix}a_{11} & a_{12} & a_{13}& \cdots & a_{1n}\\ a_{21} & a_{22} & a_{23} & & a_{2n}\\ \vdots & \vdots & \vdots & & \vdots\\ a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn}\end{pmatrix}\end{align*}\begin{pmatrix}x_1\\ x_2 \\ x_3\\ \vdots \\ x_n\end{pmatrix}=\begin{pmatrix}\sum_{j=1}^n a_{1j}x_j \\ \sum_{j=1}^n a_{2j}x_j \\ \sum_{j=1}^n a_{3j}x_j\\ \vdots \\ \sum_{j=1}^n a_{mj}x_j \end{pmatrix}$$ then (we will look at the real case, then $A^*=A^T$ that is the tranpose. $$\langle Ax,y\rangle=\sum_{i=1}^m \sum_{j=1}^n a_{ij}x_jy_i=\sum_{
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Linear Functionals Part I

  Let us look at linear functionals briefly, these are the foundation of frame theory... DEFINITION:  Let $\mathcal{V}$ be a finite-dimensional vector space. A linear functional is linear transformation $\phi : \mathcal{V}\longrightarrow \mathbb{F}$, that is a scalar-valued function that takes as input a vector. Now let us take a look at the Riesz Representation Theorem, this will require some preliminaries... First, let $A=[a_{ij}]$ be an $m\times n$ matrix over the field $\mathbb{F}$, the ADJOINT matrix $A^*$ of $A$ is the matrix $B=[b_{ij}]\in\mathbb{F}^{n\times m}$ such that $b_{ij}=\overline{a_{ji}}\ \forall \ \ i,j\in\mathbb{N}$. Simply put, the adjoint is the conjugate transpose of the matrix... PROPOSITION  Let $A,B$ be two matrices over the field $\mathbb{F}$, and let $\alpha,\beta\in\mathbb{F}$. Assume that $A,B\in\mathbb{F}^{m\times n}$, that is the same sizes. Then we have the following properties... i) $(\alpha A+\beta B)^*=\overline{a}A^*+\overline{b}B^*$ ii) $(A^*)^*=A$
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The definition of a finite frame and putting equations in blogger

Okay here is an attempt to writing an equation... I used an exchange post to modify the html Example 1: $x^2$ Example 2: $$x^2$$  This was the link I used:  https://support.google.com/blogger/thread/234344761/inserting-latex-equations?hl=en DEFINITION (FRAME)  Let $V$ be an inner product space $\{e_k\}_{k\in\mathbb{N}}$ be a set of vectors in $V$. The set $\{e_k\}_{k\in\mathbb{N}}$ is a frame  of $V$ if it satisfies the frame condition . This is, if there exists constants $A,B$ such that $0<A\leq B<\infty$ where for each $v\in V$ we have   $$A\lVert v\rVert^2\leq \sum_{k\in\mathbb{N}}|\langle v,e_k\rangle|^2\leq B\lVert v\rVert^2\ \forall \ \ v\in V$$ where $A,B$ are the frame bounds. This is the other definition I used from the book "Frames for undergraduates"... DEFINITION (FRAME)  A frame for a Hilbert space $\mathcal{H}$ is a sequence of vectors $\{x_i\}\subset\mathcal{H}$ for which there exist constant $0<A\leq B<\infty$ such that for every $x\in\mathcal{H}$
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First blog entry

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This is the first entry of the Finite Frames blog. This blog will be dedicated to showing my mathematical analysis of finite frames...more to be added.. Below is a great reference book that I will be following frequently, there are other books that I will also mention for reference...
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