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normed
normed, a. Math. (nɔːmd) [f. norm + -ed2.] Having a norm.1935 Trans. Amer. Math. Soc. XXXVIII. 360 We now add the permanent assumption that 𝔅 is ‘normed’, that is, that there is associated with 𝔅 a rule assigning to every ξ {elem} 𝔅 a number ‖ ξ‖ called the ‘norm’ of ξ, and satisfying [etc.]. 194... Oxford English Dictionary
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Normed algebra
In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: Some authors require it to have a multiplicative wikipedia.org
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normed spaces - How are norms different from absolute values ...
1 Answer. The absolute value is a particular instance of a norm. Or perhaps, you can think of norms as functions V → R where V is a vector space over a field F, and "absolute values" are "norms on the base field". The absolute value is a function | |: R → [0, ∞); given any real number r, you get a nonnegative real number that we write | r ...
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Auxiliary normed space
: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces). wikipedia.org
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Series in incomplete normed space We have known that "A normed space $X$ is a Banach space if and only if each absolutely convergent series in X converges". We would like to find an **explicitly** incomplete normed sp...
This seems to me to be a relatively simple example: $X=$ set of all real sequences with finite support (i.e., there are only finitely many non-zero elements) $\|x\|=\sup\limits_{n\in\mathbb N} |x_n|$ Consider the sequence $a_n=(0,\dots,0,\frac1{n^2},0,0,\dots)$ and the series $\sum a_n$ in $X$. This...
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Normed vector lattice
In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose Properties Every normed lattice is a locally convex vector lattice. wikipedia.org
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Is there a normed space with dense basis? Is there a normed space $(X, \lVert \cdot \rVert)$, in which one can find a (Hamel-) basis $B$ of $X$ such that $B$ is dense in $X$? And if yes: Does every normed space $X$ w...
In fact any infinite-dimensional separable normed space $X$ has a dense Hamel basis.
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A finite dimensional normed space I would like to find a short proof for the following theorems: Theorem 1. _A normed space is finite dimensional iff all of its linear functional is continuous._ Theorem 2. _A normed...
The direction you asked for in the comments: Let $X$ be an infinite dimensional normed space. 1.
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Show that any bounded linear functional on a normed linear space is continuous Show that any bounded linear functional on a normed linear space is continuous. Can we say that it is uniformly continuous ? Also, is ...
Let $L:X\to\mathbb R$ be a bounded linear functional on a normed space $X$.
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When a metric space is a normed space? I'm trying to figure out that which condition should be provided for a metric space to be normed also?
When the metric is induced from a norm. This kind of metric space $(X,d)$ must satisfy $$ d(x+a,y+a)=d(x,y)$$ $$ d(\alpha x,\alpha y)=|\alpha|d(x,y)$$ for all $x,y,a\in X$,and scalar $\alpha$. And $X$ must be a vector space.
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Closure of a nontrivial normed vector subspace that is equal to the whole space Can you show me an example of a normed vector subspace $S$ strictly included in a normed vector space $V$ whose closure is equal to the w...
Expanding on David's example: the span of the canonical basis in $\ell^p(\mathbb{N})$, $1\leq p <\infty$. Another example: $C[0,1]$ in $L^2[0,1]$.
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Proof of "Dual normed vector space is complete" < As in the introduction of dual norm by Wiki, it says dual normed space $X'$ is always complete. How to prove that? or at least explain that? We all know the normed ...
_Hint_ If $(f_n)$ is a Cauchy sequence of bounded functionals $f_n:V\to k$, take any $x$ and prove $f_n(x)$ is a Cauchy sequence in $k$. I'd take you assume your base field is complete, for example $k=\Bbb R$.
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Connectedness and normed spaces We are all familiar with the concepts of (Connectedness) , but I have a question : Should every normed space be connected ?
Yes, if we consider normed spaces $X$ to be vector spaces over $\mathbb{R}$ or $\mathbb{C}$.
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Are finite dimensional normed linear spaces locally compact? > > Let $(X,\| \cdot \|)$ be a finite dimensional normed linear space. Can we say that $(X,\| \cdot \|)$ is a locally compact normed linear space? Please ...
If $X$ is a normed vector space on $\mathbb{R}$ (or $\mathbb{C}$) then $X$ is locally compact iff it is finite dimensionnal. $\mathbb{Q}$, as seen as a $1$-dimensionnal normed vector space on $\mathbb{Q}$, is not locally compact since no neighbourhood of the origin is compact
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Examples of metric vector spaces but not normed ? Normed but not prehilbertian? What examples (non-trivial) of vector spaces are : * Normed but not prehilbertian ? * Metric but not normed ? In the schoolroom ...
For the second point, just think of a metric not invariant under traslation. For the first, you want that your norm doesn't satisfy the parallelogram law (otherwise you can construct an inner product which induce the same norm). For the first point $\ell^p$ norm are famous examples! for the second :...
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