# 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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### Proving that function is norm [duplicate]

I have a problem with proving that $$\|x\|_p=\left(\sum^{n}_{i=1}|x_{i}|^{p}\right)^{1/p}$$ is a norm where $p$ is number bigger than $1$ or $2$ the conditions are quite instant, but I can't ...
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### Differentiating norm containing vectors and a matrix

I would like to differentiate $$D = ||L^{-1} (x-y)||_2^{2}$$, while x and y are vectors and L is a matrix Can someone show me how to do this? In other words, how to calculate: $$\frac{dD}{dx}$$ ...
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### Distance between two points wrt position and orientation

I'm looking for a distance function $d: \mathbb{R}^3 \times \mathbb{R^3}\rightarrow \mathbb{R}_0^+$ between two points given by $(x, y, \varphi)_{1,2}$, where $(x,y)$ is the position in two-...
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### Norm equivalence and Banach spaces

If $(X,\|\cdot\|_1)$ is a banach space, and $\|\cdot\|_1$ is equivalent to $\|\cdot\|_2$, then $(X,\|\cdot\|_2)$ is a banach space. Does it also hold that if $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_2)$ ...
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### What is the norm on the space $L^{\infty}([0,T];L^1(\Omega))$

The title is my question, so What is the norm on the space $L^{\infty}([0,T];L^1(\Omega))$
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### Expressing continuity in terms of seminorms

Let $X$ and $Y$ be locally-convex topological vector spaces, with topologies given by families of seminorms $(p_i) _{i \in I}$ and $(q_j) _{j \in J}$, respectively. If $L : X \to Y$ is a continuous ...
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### What is a norm topology in functional analysis?

I am currently reading up about norm topology, I have a background in functional analysis but I do not know anything about topology, aside from that topology is a collection of open sets with some ...
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### Why does $1$-norm of a given vector form a diamond? [closed]

Why does $1$-norm of a given vector form a diamond? I came across this while watching a lecture on norms on youtube. Could not understand why this happens.
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### Fixed point problem with matrix

I am looking for guidance for the following fixed point problem. $A$ is an $n\times n$ matrix. The rows of $A=(A_{i})_{i\in N}$ are in the set $A_{i}\in\mathcal{A}$, where $\mathcal{A}$ is the set ...
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### sparse norm for optimization problem

I want to solve an optimization problem in general form: $$\arg \min f(x) + \lambda *g(x)$$ and i want to choose / define a $g(x)$ in a way to have a sparse solution such that between two possible ...
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### How are absolute value of a field $F$ and norm of vector space $F/F$ related?

So I think it's true that a field $F$ and its respective vector space $F/F$ are isomorphic, since they consist of the same elements, and the operations of $F$ (addition,mult.) and $F/F$ (vector ...
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### Banach Lemma misunderstanding?

The Banach Lemma: Let $B$ be an n x n matrix . If in some induced matrix norm $\|B\| < 1$, then $I + B$ is invertible and $\|(I + B)\|^{-1} ≤ \frac1{(1− \| B \|)}.$ Question: Consider the ...
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### Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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### A counterexample to $f_{n}\xrightarrow{\|\cdot\|_{L^{1}}}f$ implies $f_{n}\xrightarrow{\|\cdot\|_{C^{0}}}f$

$\|f\|_{L^{1}}\le\|f\|_{C^{0}}$ so $f_{n}\xrightarrow{\|\cdot\|_{C^{0}}}f$ implies $f_{n}\xrightarrow{\|\cdot\|_{L^{1}}}f$. I want to prove the converse is false but cannot come up with a ...
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### Show that the operator $S\in B(Y)$ (the set of bounded linear operators from $Y\to Y$)

I am having trouble with the following problem: "Suppose we have an operator $S:Y\to Y$, where, $$S(g)(y)=g(y)-\int_0^1g(x)dx$$ and where we have $Y=C([0,1])$, equipped with the uniform norm ...
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### Progressed : Convergence problem in Hilbert Space and necessity of inner product

******** PROGRESS : so thanks to Ian's great comment I can get by the proof and that completeness is necessary but I need to know does this hold for general Banach spaces that are not Hilbert spaces? ...
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### Is the max matrix norm induced?

Let $\|A \| = \max_{1 \le i,j \le n} |a_{ij}|$, where $A$ is a square matrix. I can prove that this is a matrix norm, but is it an induced norm?
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### Can someone explain why $\| \cdot \|_p, p = \frac{1}{2}$ isn't a norm?
$\| \cdot \|_p, p = \frac{1}{2}$ or "half norm" is not a norm What is a quick way to verify that it is indeed not a norm?