Questions tagged [normed-spaces]
A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.
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Approximate $L^2$ norm of sum of vectors
Suppose I need to compute the $L^2$ norm $| \cdot |$ of the sum of $n$ vectors $v_i \in \mathbb{R}^d$.
Denote $k = | \sum_{i=1}^n v_i |$, which can be achieved via adding all the vectors and then ...
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Finite Dimensional Subspace of a Normed Linear Space is complete.
I am working through the wiki proof that every finite dimensional subspace of a normed linear space is closed. The proof goes as follows:
Let $(V ,\| \cdot \|)$ be a normed linear space. Let $W$ be a ...
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Space of functions of infinitely many variables with norm different from the uniform one
Consider a family of compact metric spaces $(\Omega_t)_{t\in\mathbb{R}}$, and the corresponding product space $\Omega=\prod_{t\in\mathbb{R}}\Omega_t$. I will write its generic element as $\omega=(\...
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Can a point be closer to all the vertices of a convex polytope than another point inside that polytope?
Consider a set $X = \{x_i \in \mathbb{R}^n\}$ and denote its convex hull
$$
C \equiv \bigg\{ \sum_i \lambda_i x_i : \lambda_i \geq 0 \text{ for all } i \text{ and } \sum_i \lambda_i = 1 \bigg\}.
$$
I ...
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Show a function is not a norm on $\mathbb{R}^2$ by failing the triangle inequality
I'm trying to show the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(v) = (\sqrt{|v_1|} + \sqrt{|v_2|}$ is not a norm on $\mathbb{R}^2$ by showing the triangle inequality doesn't hold. That ...
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Is the duality product of dual spaces unique?
Let $\Omega \subset \mathbb{R}^n$. Consider, for example, the Sobolev space $H^1_0(\Omega)$. It is known that the dual is $H^{-1}(\Omega)$ and is Banach with respect to the operator norm
$$||f||_{-1} =...
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Mock scalar product's nonlinearity as a derivation
Context. I was reading about the converse of the parallelogram law (a norm satisfying the law defines an inner product), and in an answer on MO it is stated that this is impossible without the norm's ...
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Showing that any norm is Lipschitz
I am asked to prove that in a normed space the norm is Lipschitz. I have seen concrete examples but never something general. If one has a norm $||x||$, the distance is induced by $d(x,y) = ||x-y||$. ...
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Sets invariant under isometries of $\mathbb{R}^n$
Let $f : \mathbb{R}^n \to \mathbb{R}^{n}$ be an isometry. Is it true that $f(U) \subseteq U$ implies that $U \subseteq f(U)$ for an arbitrary subset $U$ of $\mathbb{R}^n$?
It is intuitively clear to ...
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Prove that if $f_n$ is bounded, there exists $M>0$ such that $\Big| |f_n| -|f_n -f| -|f|\Big| \le M$
Let $(f_n)_n\subset \mathbb R$ be a bounded sequence of real valued functions. Moreover, assume that $f_n\to f$ a.e.
I would like to prove that there exists $M>0$ such that
$$\Big| |f_n| -|f_n -f| -...
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Where does the $\ell_2$ norm subdifferential come from?
Looking at the definition of the subdifferential, we have that $v$ is a subdifferential of a function $f$ at a point $x$ if
$$ f(y) \geq f(x) + g^T(y-x), \forall y$$
Now, for $f(x) = \|x\|_2$, it's ...
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Showing operator norm over $\mathbb{R}$ attains supremum
Let $V, W$ be finite dimensional normed vector spaces over the real numbers. Define the operator norm on the vector space of linear maps $L(V,W)$ as $||A|| := \sup \{ ||Av|| \mid v \in V, ||v||=1 \}$.
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$L^2$-ness of the finite difference operator
Is there a proof of the fact that the operator $\mathfrak{D}$ defined by $\mathfrak{D}[f](x,y):=\dfrac{f(x)-f(y)}{x-y}$ is continuous from $H^n(\mathbb{R})$ to $H^m(\mathbb{R}^2)$ or even $L^2(\mathbb{...
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Removing Finite Measure Assumption in the Proof that 𝐿𝑝(𝜇)⊂𝐿𝑞(𝜇) where 1⩽𝑞<𝑝<∞
Looking at this question and the corresponding answer from @DanielFischer:
$L^2$ function on finite interval implies $L^1$?
I was trying to see if we really need the finite measure assumption ($\mu(X) ...
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How can a non-reflexive space have a reflexive subspace?
I know that this can happen (take any one-dimensional subspace, for instance), but I had the following thought while reading Kadison and Ringrose (Fundamentals of the Theory of Operator Algebras, Vol ...
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Counterexample to: "Every bounded sequence in a normed space has an accumulation point"
Every bounded sequence in a normed space has an accumulation point (?)
I am trying to find a counterexample to show that not every bounded sequence in a normed space has an accumulation point, as I ...
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A characterization of closure of a certain class of sets in $\mathbb{R}^n$
Consider a set $K\subset \mathbb{R}^n$ that is symmetric ($B = -B$) and verifies $aK\subset bK$ if $|a|<|b|$. Can I conclude that $\overline{K} = \cap_{a>1}aK$? If not, does the result hold ...
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Distance that comes from a norm
In a certain proposition, I am given a distance function in a vector space ($d : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{V}$) that satisfies the following conditions:
$\forall \lambda \in \...
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2-norm of transpose proof
I don't understand the proof of ‖x‖2=‖xT‖2.
I know you can use the Cauchy-Schwartz inequality and that you can prove it by first showing ‖xT‖2 ⩽ ‖x‖2 and then ‖x‖2 ⩽ ‖xT‖2. I get proving both ...
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Minimax and Maximin
I am reading about Q-Learning and I come across these minimax and maximin equations, where they relate it to the sup-norm.
$$
-\|Q\|_{\infty}\leq \min_{s}\max_{b}Q(s,b)\qquad\text{ and }\qquad \max_{s}...
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Intuition behind the Euclidean orthogonal projection
Let $A$ be a closed and convex set in $\mathbb{R}^{n}$, let $x_{0}\in\mathbb{R}^{n}\setminus A$, and define $f:\mathbb{R}^{n}\longrightarrow (0,\infty)$ by $f(x):=\frac{1}{2}\|x-x_{0}\|^{2},$ where $\|...
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Show that $T$ is continuous and bijective if $\inf\limits_{k \in \mathbb N}\ \{|\alpha_k|\} \gt 0.$
Let $\{\alpha_k\}_{k \geq 1}$ be a sequence of real numbers such that $\inf\limits_{k \in \mathbb N}\ \{|\alpha_k|\} \gt 0.$ Consider the linear operator $T : \ell^2 \longrightarrow \ell^2$ defined by ...
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Conections between Parseval's Identities
I know how to prove that if $f,g \in \mathcal{S}$ (Schwartz space - or even ${L}^2$), then $$f*g \in \mathcal{S}$$ and $$\|f\|_2^2=\|\widehat{f}\|_2^2.$$ Here, $$\|f\|_2=\sqrt{<f,f>}= \left[ \...
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Expected value of a euclidean 2 space norm [closed]
Say we have $A=(a_1,a_2)$ and $B=(b_1,b_2)$ which are independent and uniform on say $[0,x]^2$ how would we find $\mathbb{E}(||A-B||^2)= \mathbb{E}((a_1-b_1)^2+(a_2-b_2)^2)$.
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Show the continuity of a function between vector space
I consider the function defined by $\phi(f)$ which associates to each function $f$ from $C([0,1],\mathbb{R})$ the function $\phi(f)(x) = \int_{0}^{x} tf(t)dt$ from $[0,1]$ to $\mathbb{R}$.
My idea is ...
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Proof by contradiction that $(\ell^p)^* \subseteq \ell^q$
For $p \in (1, \infty)$, let $\ell^p = \{(x_n) : \sum |x_n|^p < \infty\}$ and let $(\ell^p)^*$ denote the dual space of bounded linear forms on $\ell^p$ $(\mathbb{F} = \mathbb{R})$.
It is ...
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Squared $L_2$ norm is strongly convex w.r.t. $L_1$ norm
I am not assured to say True or False of the statement written in the title.
Suppose a function $h$ receives a $K$-dimensional vector $\boldsymbol{p}$, which is a member of $K$-simplex (i.e., $\...
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Minimax and Infinity norm
I am reading a paper on Q-learning with the state space and action space denoted as $\mathcal{S}$ and $\mathcal{A}$ respectively. Under a certain policy $\pi$, the value function and Q-function is ...
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Convergence of sequence vs NOT equivalent norms
Is there a pair of norms such that they are not equivalent, but for any sequence of $(x_k)$, we have $\left \| x_k-x \right \|_1\to 0$ if and only if $\left \| x_k-x \right \|_2\rightarrow 0$?
I know ...
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the space $C((0,\infty,L^{2}(\omega))$ and $C((0,T,L^{2}(\omega))$
Hi sorry if it seems evident but i need a help for these statements
1)$N(u)=\sup_{t\in R^{+}}\|u(t)\|_{L^{2}(\omega)}$it is not a norm?and if it is not which norm i need to choose for space $C((0,\...
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L2 Norm of Pseudo-Inverse Relation with Minimum Singular Value formal source?
Given a full column rank matrix $A \in \mathbb{R}^{n\times m}$. The left pseudo-inverse is $A_{\text{left}}^{-1}=(A^\top A)^{-1}A^{\top}$. Then we have the following relationship
$$\|A_{\text{left}}^{-...
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Why do we use this norm on sequence spaces
I'm studying sequence spaces $\ell^p=\{(x_j)_{j\in \mathbf{N}}:\sum_{j\in \mathbf{N}}|x_j|^p<\infty\}$ for $1\leq p<\infty$.
This is a vector space (I'm not sure how to prove it is closed under ...
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Doubt in proof of Jordan - von Neumann theorem
In Paliogiannis and Martin A. Moskowitz's Functions Of Several Real Variables, the Jordan - von Neumann theorem is given as follows:
The inner product is defined as: $$\langle x,y \rangle = \frac{||x+...
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Does the space of positive signed measures have an interior
Let $\mathbf{X}$ be a compact interval on the real line with $B(\mathbf{X})$ representing the Borel algebra of $\bf{X}$. Let $M(\mathbf{X})$ be the space of finite signed measures, and let $\Vert\cdot\...
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Is there a geometrical reason because we define a norm as a map from a space into $[0, +\infty[$ and not into $[0, +\infty]$?
For convenience i replace a precise definition of a norm:
Def: Let X a vector space on a subfield $\mathbb{F}$ of $\mathbb{C}$,
a norm on X is a map $p:\mathbf{X} \rightarrow \mathbb{R}^+=[0, +\infty[$...
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Trying to find operator norm
Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces and let $V=B(\mathcal{H}, \mathcal{K})$. Fixed $T_i \in V$ for $i=1, 2$.
Define $T: V \oplus V \to V \oplus V$ as $$T( v_1, v_2) = (T_1T_2^*v_1, ...
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Operatornorm of powers of Matricies with integer coefficient
Let $A\in GL_n(\mathbb{Z})$ have infinite order, so $A^k\neq Id_n$ for all $k>0$. The operator norm is defined by $\lVert A \rVert=\max\{\lVert Av\rVert \mid v\in\mathbb{R}^n: \lVert v\rVert=1\}$. ...
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Clarification on a remark by Schaefer on nuclear spaces and absolute versus unconditional summability
In my (poor) attempt to answer Unconditional and absolute convergence in non-Banach spaces, essentially asking about whether or not we can get rid of the completeness assumption in the Dvoretzky-...
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Unconditional and absolute convergence in non-Banach spaces
I know that, by Dvoretzky-Rogers theorem, we know that in a Banach space $X$ the following are equivalent:
$X$ is of finite dimension.
Every unconditonally convergent series is absolutely convergent.
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Relationship between trace norm (a.k.a. Schatten-1 norm) of a matrix and the vector norm of the matrix's row average?
I'm trying to understand whether a connection exists between two seemingly different optimization problems in machine learning.
Setup: Suppose I have $N$ points $x_1, ..., x_N \in \mathbb{R}^D$, where ...
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When is the Minkowski functional a norm?
Let $X$ be a real or complex topological vector space and let $A\subseteq X$ be absorbing, balanced, and convex. Then the Minkowski functional $\mu_A$ is a seminorm. Can we add some conditions that ...
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Inequality between $L^p$ norm and Schwartz seminorm
Let $u\in \mathscr{S}$ (Schwartz space). I can prove easily that $u$ is in $L^p$ for every $p$. But I cannot prove the following. Let $\{p_k\}$ be the family of seminorms that defines the topology of ...
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Find an $\mathsf{L}^{p}$ space for financial markets with infinitely many risky assets
Given a sequence of random variables $\mathbf{x}=(X_{1},X_{2},\cdots
,X_{n},\cdots)$, with $X_{i}\in\mathsf{L}^{2}(\Omega,\mathcal{F},\mathrm{P})$
for each $i$ such that $\left\Vert X_{i}\right\Vert =\...
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Define $\ell:C[-1,1]\to\mathbb{R}$ such that $\ell(f)=\int_{-1}^0f-\int_0^1 f$ and let $\|f\|=\max_{t\in[-1,1]}|f(t)|$. Is $\|\ell\|=2$?
$\newcommand{\R}{\mathbb{R}}$
I'm doing a practice exam for Real Analysis and am wondering about this specific question:
Let $C[-1,1]$ be the set of all real-valued continuous functions on $[-1,1]$. ...
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Prove that two norms are equivalent in the space of polynomials of degree less than or equal to 2
Let $\mathbf{P}$ The set of polynomials of degree $\leq$ two, with real coefficients in $[0, 1]$, if $l=ax^2+bx+c$ in $\mathbf{P}$, we define the norms $||l||_{1}=|a|+|b|+|c|$ and $||l||_{\infty}=\...
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Example of a sequence that is Cauchy in a stronger norm and convergent in a weaker norm, but not convergent in the stronger norm?
A norm $\|\cdot\|_1$ on a normed vector space is called stronger than $\|\cdot\|_2$ when $\|x\|_2\leq M\|x\|_1$ for some $M>0$ and all $x$. It is a standard trick (e.g. in proving completeness) to ...
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Is there an infinite-dimensional normed vector space in which the set complement of a bounded set may have more than one unbounded component?
Let $n \ge 2$. Using the fact that $\{x \in \mathbf R^n : \lVert x\rVert > R\}$ is connected for each $R > 0$, we can show that if $B \subseteq \mathbf R^n$ is bounded, then $\mathbf R^n \...
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confused in the choice of norms
i want to prove that the followinf function is bijection:
$$A: H_{0}^{1}(\Omega)\cap H^{2}(\Omega) \to L^{2}(\Omega)$$
$\Omega$ is $C^{1}$ bounded domain of $R^{n}$
and the hint was to use lax ...
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Closure of balanced convex set and its dilation
While reading a proof, one of the steps was showing that for a (specific) balanced and convex set in a Banach space we had the inclusion $\overline{B} \subseteq 2B$. The proof continued using specific ...
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Banach space which has Radon-Nikodym Property +Weakly sequentially complete but not reflexive.
I have been trying to understand reflexivity of a Banach space from geometric point of view. I know that a reflexive space has Radon-Nikodym Property (RNP) and is Weakly sequentially complete(WSC) by ...
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