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

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Derivative of function that includes norm

I was solving the problem: find the derivative of a function f : H → R, $f (x) = \sin ||x||^3$ (H is Hilbert space). I got the answer $f'(x)=3\cos||x||^3 x||x||$. Is this correct or I am doing ...
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Any two norms on finite dimensional space are equivalent

Any two norms on a finite dimensional linear space are equivalent. Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for ...
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77 views

Show that every operator norm is consistent

Is the following a correct way to show that operator norms are consistent? $$ \|AB\|=\max_{Bx \ne 0}\frac{\|ABx\|_\alpha }{\|x\|_\alpha} =\max_{ Bx\ne 0}\frac{\|ABx\|_\alpha}{\|Bx\|_\alpha} ...
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100 views

Abstract Algebra: Field extensions

I'm trying to prove the following but it causes me a lot of trouble: Let $L$ be a finite extension of degree $n$ of a field $K$ with characteristic $0$. Let $\sigma_1,\dots,\sigma_n$ be different $K$ ...
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200 views

Proof that two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent

Two norms $\def\norm#1{\lVert#1\rVert}\norm\cdot_1$ and $\norm\cdot_2$ are equivalent iff $\;\exists\;c_1,c_2>0$ such that $c_1\norm x_1\le \norm x_2\le c_2\norm x_1$ Show that $\norm ...
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75 views

Minimization of norms

How do I minimize the following? $ min_{z>0} - zt + 1/2\ z\ ||\ Y + X_k\ /\ z\ ||_2^2 $ Also, $X_k^TX_k = 1 \ \ \forall k $ I am given that the answer should be : $ \sqrt{Y^T - 2t} + Y^TX$ ...
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37 views

What could be the lead to prove $||X||_2 \leq ||X||_F \leq \sqrt{rank(X)}||X||_2$?

In the above statement, $||X||_2$ = $L_2$ norm of X and $||X||_F$ = $Frobenius$ norm of X. It appears to me that the $L2$ norm of X and $Frobenius$ norm of X are the same. How should i proceed to ...
2
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50 views

Help showing $\phi _k$ is a bounded linear functional

Let $V$ be the space of continuous functions on the interval $[-\pi , \pi]$ with the $L^2$ norm $$\lVert f\rVert_2=\left(\int_{-\pi}^\pi |f(t)|^2\mathrm dt)\right)^\frac{1}{2}$$ For $f$ in $V$, define ...
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103 views

How to force unitary Euclidean norm in a complex matrix by multiplication with a diagonal matrix

I need to solve the following problem: Suppose a non-sparse, non-singular complex matrix $\mathbf{P}$. If I want to force all rows in $\mathbf{P}$ to present unitary Euclidean norms by multiplying ...
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90 views

Calculating the norm of $S\otimes T$ for bounded linear $S,T$.

Let $X,Y,W,Z$ be Banach Spaces. Let $X\otimes_{\pi}Y$ denote the tensor product endowed with the projective norm. If I have $S\in B(X,Z)$, $T\in B(Y,W)$, it is straightforward to show that $S\otimes ...
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509 views

Difference between convergence in norm, point-wise and uniform convergence

I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, ...
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1k views

Constrained infinity norm minimization

I have a problem like this: $$\min_x |Ax|_\infty \text{ s.t. } \sum_i x_i = c$$ That is, I want to find the vector $x$ whose elements sum to a constant $c$ that minimized the infinity norm of $Ax$. ...
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96 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
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51 views

Regularity and the Varitational Inequality

Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
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114 views

Operator norm estimate

Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is ...
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131 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
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149 views

finding operator norm $T_N$

How do find the operator norm of , $T_N\colon c_0\to \Bbb R$ given by $T_N(y):=\sum_{j=1}^Nx_jy_j$ ,when $N$ is a integer . TIA
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313 views

Invertible operator norm bound

Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...
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961 views

Norm equivalence Sobolev space

I have this problem: Let $k>0$ (integer) and $1 \leq p < \infty$. Show that the norms $$ ||u||_{W^{k,p}(U)} = \bigg( \sum_{|\alpha|\leq k}||D^{\alpha}u||_{L^{p}(U)}^{p}\bigg)^{\frac{1}{p}} $$ ...
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183 views

Is the polar set of convex Polytope also Polytope

Let $P$ be a convex polytope. How can I prove that the polar set of $P$ (lets call it $P^*$) is polytope? where $P^*=\{x\in\mathcal R^n:\forall v\in P, |\langle x,v\rangle|\le1\}$ . $Thanks$
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What are the maximumnorm and supremumnorm of a vector when having a basis?

I have a perhaps stupid question. When having a finite-dimensional Vectorspace $X$ (f.e. n-dimensional) and when knowing a basis $V=\left\{v_1,...,v_n\right\}$ of it, so any $x\in X$ can be written as ...
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59 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
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convergence of $L^p $ norm [duplicate]

Possible Duplicate: Limit of $L^p$ norm If I define $|f|_{L^\infty}= \lim_{n\to \infty} |f|_{L^n}$. How can I prove that this limit is esssup $|f|$?
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277 views

Quotient norm on $X\backslash M$

I have $X=(C([0,1]),||.||_1)$ where $||f||_1=\int_{0}^{1}|f(t)|dt$ and $M=\{f\in C([a,b]): f(0)=0\}$. Now I have three questions: 1) Is the quotient norm a norm on the quotient space X\M ? What I ...
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658 views

Does the limit of a convergent sequence depend on the norm?

Let $X$ be a vector space, and $\|\cdot\|_1$ and $\|\cdot\|_2$ two different (non-equivalent) Norms on $X.$ Let $(x_n)\subset X$ be a sequence and $x\in X$ such that $\lim_{n\to\infty}\|x_n-x\|_1=0.$ ...
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583 views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...
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How to prove these three norm equivalence problems

Given $A \in R^{m\times n}$, I have these three norm equivalence equations: $\|A\|_2 \le \|A\|_F \le \sqrt {n}\|A\|_2$ $\frac {1} {\sqrt n}\|A\|_{\infty} \le \|A\|_2 \le \sqrt {m} \|A\|_{\infty}$ ...
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350 views

Need help with relative and absolute errors?

Lets assume I have $Ax=b$ equation, where $A$ is $2$x$2$ matrix. 1) I want to find an A, x, and b such that relative error in x is small but absolute error in x is large 2) Also want to find A, x, ...
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Equivalence between $Lip \ norm$ and $C_1 \ norm$.

Let $f\in C^1([a,b])$. Prove that $\|f\|_{C^1} = \|f\|_{Lip}$. By definition of Lip norm and $C^1$ norm, it is equivalent to prove that $\|f'\|_{\infty}=Lip(f,(a,b))$, where the second member is the ...
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299 views

inequality between norms of vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. It is known that $\|a\|_{\infty}\leq \|a\|_2\leq \sqrt n\|a\|_{\infty}$. Let $k<<n$. For which kind of vectors the following would be true: $$ ...
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133 views

What does this norm $|| \cdot ||_{{C^0} (K) }$ mean?

$$ || A ||_{C^0 (K)} $$ Here $A$ is $ n \times n $ Hermitian, Positive definite matrix, and $K \in \mathbb R^n$.
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How do you expand this frobenius form?

I'm trying to expand this Frobenius form $||C \circ (A-XB)||_F^2$ (here $\circ$ is the Hadamard point-wise multiplication). I want to find the minimum value with respect to X. $$ ...
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A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
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125 views

Example of a vector norm for which $\|I\|<1$

In order to prove a larger assumption, I need to find a vector norm over $M_n$ such that $\|I\| < 1$. None of the standard $p$-norms, nor the infinity norm work. I know that for matrix norms, ...
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383 views

Matrix Norm Identity Derivation

I am having trouble figuring out where something in a book I am reading is coming from. (The book is Matrix Computations by Golub and Van Loan, 3rd edition, p.58.) It will probably be obvious once ...
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15 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
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How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
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Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
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Find Solution regarding 2-Norm

I try to understand that, but I have no clue what do to and how to do it. $A$ is a $m \times n$ matrix with $rg(A)=m$. Find the solution for $Ax = b$, which is regarding to the $2$-norm (I guess ...
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Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
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How to prove that $\|x+y\|_p<\|x\|_p+\|y\|_p$? [duplicate]

Let $l_p=\{(x_n)\in\mathbb{R}^\mathbb{N}: \sum_{n=0}^\infty |x_n|^p<\infty\}$ and consider the following norm in $l_p$: $$\|x\|_p=\left(\sum_{n=0}^\infty|x_n|^p\right)^{1/p}$$ for ...
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a norm is symmetric if and only if it is unitarily invariant

how can I prove this : A norm on $\mathcal{M}_n(\mathbb{C})$ is symmetric if and only if it is unitarily invariant ? My attempt I know that a symmetric norm is a norm which verifies : $$N(ABC)\leq ...
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24 views

Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i ...
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vector space of continuously differentiable functions is complete regarding a specific norm

Consider $C^1[a, b] = \{f: [a, b] \to \mathbb{C}\mid f\text{ continuously differentiable}\}$. Consider the following norm: $$\|f\|_{C^1} = \|f\|_\infty + \|f'\|_\infty$$ Now, it needs to be shown ...
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Prove the set is closed with respect to its norm…

Let $V$ be a normed vector space over R. Let $W$ be a proper closed subspace of $V$. We say $w^*$ is a best approximation in $W$ to $v^* \in V$ if $\|v^*-w^*\| \leq \|v^*-w\|$ for all $w \in W$. ...
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Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
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28 views

Why doesn't the $L_2$ norm differentiable at $x=0$?

Why doesn't the $L_2$ norm differentiable at $x=0$? Let's define $N(x)$ as the norm function. I know that for every $x\ne 0$: $$\frac{\partial N}{\partial x_i}(x) = \frac{x_i}{\|x\|}$$ What ...
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34 views

Equivalence of norms between two Banach spaces

I just learnt open mapping theorem. And I met a statement online asserting that If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent. Can we ...
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37 views

Norm of Fredholm operator in $L^1$

Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by $$ Tf(x)=\int_0^1 k(x,y)f(y)\, dy $$ where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find ...
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Proof for Norms in Vector Spaces

Prove that if a norm $\|x\|$ on a real vector space satisfies the parallelogram law, then the polarization identity defines an inner product and that the norm associated with this inner product is the ...