# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### Prove this integral inequality

Prove this assuming $f$ is integratable: $$\int_{-\pi}^\pi\vert f(t)\vert \, dt\leq \sqrt{2\pi}\sqrt{\int_{-\pi}^\pi\vert f(t)\vert^2}\, dt =2\pi \Vert f\Vert.$$ I tried to square both sides and use ...
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### Why is this vector of unit length?

Given $A$ a symmetric positive definite real matrix and $V$ a non zero column vector define: $B=V^\top A^{-1} V$ and since $A$ is p.d., $B>0$. Also, $A$ has a unique square-root $\sqrt{A}$. I ...
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### the norm a functional that maps a convergent sequence to its limit

Given the linear space $\vec{E}$ of real convergent sequences, dotated with the norm \begin{equation*} \begin{aligned} \| \cdot \| : \vec{E}&\to \mathbb{R} \\ x=(x_n)_{n\in\mathbb{N}} ...
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### Prove an inequality involving a norm

We define the following inner product on intergrable, $2\pi$ periodic functions from $\mathbb{R}$ to $\mathbb{C}$: $$\langle f,g\rangle = \frac{1}{2\pi} \int_{-\pi}^\pi f(t)\overline{g(t)}\ dt$$ I ...
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### If $V$ is a finite dimensional with two norms then $\Vert v\Vert_1 \leq c\Vert v\Vert_2$

Suppose $V$ is finite-dimensional and $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on $V$ with corresponding norms $\Vert \cdot\Vert_1$ and $\Vert \cdot\Vert_2$. ...
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### Hilbert space sequence

Let $$L^2:=\left\{f:[0,1] \to \mathbb R \; \middle| \; \int_0^1 |f|^2 < \infty\right\},$$ $$\langle f,g \rangle:= \int_0^1 f g,$$ and $$\|f\| := \sqrt{\langle f, f \rangle}.$$ Prove that ...
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### Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
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### Find the norm of the following operator.

Take $X=C([0,1])$ with the uniform norm, $\|f\|=\sup_{x\in[0,1]}|f(x)|$, and define the operator $T:X\to X$ by, $$T(f)(x)=f(x)-\int_0^1f(s)ds$$ Find $\|T\|$. I was hoping to solve this ...
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### Randomly select vector with bounded norm

I would like to select randomly an $n$-dimensional real vector $\mathbf{x}=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ such that its norm is bounded by a positive real number, say $\sqrt{a}$, $a\in\Bbb{R}_+$. ...
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### Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?

I am currently working on showing that $\Vert Px \Vert_{2} \leq \Vert x \Vert_{2}$, where $x \in$ an inner product space $X$, and $P$ is the orthogonal projection operator. Also, I am supposing that ...
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### Prove Operator Norm is a Norm on linear space [duplicate]

Prove that the operator norm defined by $$\left \| A \right \| = \left \| A \right \|_{V\rightarrow W} = \sup_{0\neq v\in V} \frac{\left \| Av \right \|_{W}}{\left \| v \right \|_{V}}$$ (Given norms ...
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### Convergence of sequence of discrete measures on $\mathbb{T}$ in total variation norm

Let $\mathbb{T}=\mathbb{R/Z}$ be the circle. Prove that the space of discrete measures on $\mathbb{T}$ is closed under convergence in total-variation norm in the set of measures on $\mathbb{T}$. ...
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### Equivalence of norms on $W^{1, p}(I)$

Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev ...
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### Measure theory , Functional calculus, Self Adoint

In $$L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. I need to show $\langle g_a|g_b \rangle=0$ if ...
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### Defining a distance between images

Let's consider raster images represented by bi dimensional real matrices. I have an original image $M_0$, and after transforming it several times I get a set of related images $M_n$, which have the ...
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### Compute of two norms of a function of three variables

Let $f$ be a function defined on $\mathbb R^3$ by $$f(x,y,z)=\exp(-2\mathbb i\pi (x+y+z)) |x|^{1-k} |y|^{k-1} \operatorname{sign}(x) \operatorname{sign}(z),$$ where $sign(x)$ means the sign of $x$ and ...
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### Find vectors x and y with given norms.

I've spent many hours on this and I just can't understand how to do this. Could you please go through this with me? I have a test, and I really need to understand how to do these types of problems. ...
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### What is the norm of matrices? Is it related to the norms of linear transformations? [closed]

What are the norms of a matrix? Is there any relation with norm of linear operators/transformations?
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### Equivalency of two norms

Let $U$ be a normed vector space with two norms: $|| . ||,|| . ||^{'}$ For every sequence $\{x_n \}$ that $||x_n-x||\rightarrow 0$ & $||x_n-y||^{'}\rightarrow 0$,we can conclude $x=y$. ...
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### the relation between cardinality, L1-norm and L2-norm of a vector

For every $u\in \mathbb{R}^n$, $\textbf{Card}(u)=q$ implies ${\lVert u \rVert}_1 \leq \sqrt{q} {\lVert u \rVert}_2$ where $\textbf{Card}(u)$ is the number of non-zero element (so the L0-norm). Why ...
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### Is $|(v,\frac{Pv}{||Pv||})|=||Pv||$ when $P$ is an orthogonal projection?

Suppose $P$ is an $k \times k$ matrix that represents an orthogonal projection. Let $v$ be an $k \times 1$ vector. Let the operator $(\cdot,\cdot)$ represents the scalar product. Does this ...
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### Correct this solution to finding $||A||$ of $(Af)(x) = g(x)f(x)$.

I need some help with a question I tried to solve it, but I am just not quite sure if my answer is correct. (I have got the feeling it can be - much - better). Suppose we have a complex Hilbert space ...
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### Matrix norm inequality proof: inverse of two p.s.d matrices sum

I wonder if the following matrix norm inequality holds: Let $A$ and $B$ are both strictly symmetric positive definite matrix $\|(A+B)^{-1}\|_2\leq \|A^{-1}\|_2$ ? Thanks in advance.
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### Relation of the upper triangular factor and the original matrix

Suppose $$PA = LU$$ is the LU factorization(exact) of the square real matrix A, L is the unit lower triangular matrix. Is there a way to determine the relation between the norm of $U$ and the norm ...
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Let $0 < a < b$ and $T\colon L^\infty((0,1)\times (a,b)) \to L^\infty((0,1)\times (a,b))$ be the operator defined by $$Tf(x,y) = \begin{cases}f(x+\frac yb,y), &0<x<1-\frac yb,\\\frac ... 0answers 14 views ### Write the norm ||R^h v|| just in function of v if R=[ Q \, \mathbf{0}] and the columns of Q form an orthonormal basis Let Q be an m \times (m-k) (complex) matrix where its columns form an orthonormal basis (m-k vectors). We define matrix R=[ Q \, \, \mathbf{0}], where \mathbf{0} is the m \times k zero ... 1answer 26 views ### Convergence of measures on \mathbb{T} Denote by M(\mathbb T) the set of complex-values measures on the circle \mathbb{T}=\mathbb{R}/\mathbb{Z}.Prove that D(T), the set of discrete measures on \mathbb{T} is: closed in ... 1answer 15 views ### comprehensive overview of techniques for finding the difference between vectors There are many techniques for finding the difference between two vectors. for example: the norm of the difference. absolute value of the difference Mahalanobis distance Bhattacharyya distance etc ... 1answer 39 views ### Interchange of \ell^r and L^p-norm Let (f_i)_{i\in\mathbb{N}} be a sequence of L^p-functions. What is the relation between \Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p} and \Vert \left(\Vert ... 1answer 31 views ### Source/explanation for this matrix inequality Here it is:$$z^\top M^{-1} M^{-1}z \le \|M^{-1}\| z^\top M^{-1} z.$$Where \pmb M is positive definite symmetric, z is a vector in \mathbb{R}^p (not necessarily normed!) and ||\pmb M|| is ... 1answer 32 views ### Open mapping theorem for normed abelian groups A norm on an abelian group is a function valued in \mathbb{R}_{\geq 0} which satisfies |x|=0 \Leftrightarrow x=0, |{-}x|=|x|, and |x+y| \leq |x|+|y|, not necessarily |z x| = |z| |x| for ... 2answers 38 views ### Exchanging limits with norms and linear functionals In a normed vector space X, when can we say: \lim\|x_n\|=\|\lim x_n\| and further, if f\in X^{*}, when can we say: \lim fx_n=f(\lim x_n)? 1answer 48 views ### Show that this matrix product is bounded Suppose we have a symmetric real matrix \pmb M satisfying:$$\underset{\pmb\alpha\in\mathbb{R}^p:||\alpha||=1}{\min}\;\pmb \alpha\pmb M\pmb \alpha^{\top}\geqslant k>0. Then, I am trying to ...
Stumbled upon this one in a textbook: Let there be a linear transformation $T:V\rightarrow V$ over a finite inner product space $V$. It is known that $TT^* = 7T - 12I$. How can it be proved that ...