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.

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to\mathbb R_+$ is a normed space if the three conditions are fulfilled:

  1. $\lVert x\rVert =0\Rightarrow x=0$,
  2. For all $x\in E$ and for all $\lambda\in\mathbb R$ (or $\mathbb C$), $\lVert\lambda x\rVert=|\lambda |\lVert x\rVert$,
  3. For all $x,y\in E$, $\lVert x+y\rVert\leq\lVert x\rVert+\lVert y\rVert$.

For instance, in $\mathbb{R}^n$ each of the following functions is a norm:

  1. $\displaystyle\bigl\|(x_1,x_2,\ldots,x_n)\bigr\|_2=\sqrt{{x_1}^2+{x_2}^2+\cdots+{x_n}^2}$;
  2. $\displaystyle\bigl\|(x_1,x_2,\ldots,x_n)\bigr\|_1=|x_1|+|x_2|+\cdots+|x_n|$;
  3. $\displaystyle\bigl\|(x_1,x_2,\ldots,x_n)\bigr\|_\infty=\max${$|x_1|,|x_2|,\ldots,|x_n|$}.
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Norms Induced by Inner Products and the Parallelogram Law

Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot…
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Relations between p norms

The $p$-norm on $\mathbb R^n$ is given by $\|x\|_{p}=\big(\sum_{k=1}^n |x_{k}|^p\big)^{1/p}$. For $0 < p < q$ it can be shown that $\|x\|_p\geq\|x\|_q$ (1, 2). It appears that in $\mathbb{R}^n$ a number of opposite inequalities can also be obtained.…
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Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making things more concrete: I am esp. interested in the…
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Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric on a linear space can be induced by norm? I know…
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Equivalent Definitions of the Operator Norm

How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &=\sup\{ \lVert Av\rVert\;\colon\;…
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Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
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Gradient of 2-norm squared

Could someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$? $$\nabla\|x\|_2^2 = 2x$$
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What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are identical, then I suppose the only difference between them…
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Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$

Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? A related question is: Can we proved…
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"Every linear mapping on a finite dimensional space is continuous"

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector spaces (not necessarily the same), as along as the domain…
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Why is $L^{\infty}$ not separable?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$? Our teacher gave us some hints that there exists an uncountable subset such that the distance of…
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Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it to solve problems, but in reality I still lack…
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Every proper subspace of a normed vector space has empty interior

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric $||\cdot||$). I have a proof for the case $V$ is…
Somabha Mukherjee
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On the norm of a quotient of a Banach space.

Let $E$ be a Banach space and $F$ a closed subspace. It is well known that the quotient space $E/F$ is also a Banach space with respect to the norm $$ \left\Vert x+F\right\Vert_{E/F}=\inf\{\left\Vert y\right\Vert_E\mid y\in x+F\}. $$ Unfortunately…
Jyrki Lahtonen
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Proof that every finite dimensional normed vector space is complete

Can you read my proof and tell me if it's correct? Thanks. Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t.…
Rudy the Reindeer
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