# 1.3 Normed vector spaces

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While vector spaces have additional structure compared to a metric space, a general vector space has no notion of “length” or “distance.”

## Definition 1

Let $V$ be a vector space over $K$ . A norm is a function $∥·∥:V\to \mathbb{R}$ such that

• $∥x∥\ge 0\forall x\in V$
• $∥x∥=0$ iff $x=0$
• $∥\alpha ,x∥=|\alpha |∥x∥\forall x\in V$ , $\alpha \in K$
• $∥x,+,y∥\le ∥x∥+∥y∥\forall x,y\in V$

A vector space together with a norm is called a normed vector space (or normed linear space ).

• $V={\mathbb{R}}^{N}$ : ${∥x∥}_{2}=\sqrt{{\sum }_{i=1}^{N}{|{x}_{i}|}^{2}}$
• $V={\mathbb{R}}^{N}$ : ${∥x∥}_{1}={\sum }_{i=1}^{N}|{x}_{i}|$ (“Taxicab”/“Manhattan” norm)
• $V={\mathbb{R}}^{N}$ : ${∥x∥}_{\infty }=\underset{i=1,...,N}{max}|{x}_{i}|$
• $V={L}_{p}\left[a,b\right]$ , $p\in \left[1,\infty \right)$ : ${∥x,\left(,t,\right)∥}_{p}={\left({\int }_{a}^{b},{|x\left(t\right)|}^{p},d,t\right)}^{1/p}$ (The notation ${L}_{p}\left[a,b\right]$ denotes the set of all functions defined on the interval $\left[a,b\right]$ such that this norm exists, i.e., ${\parallel x\left(t\right)\parallel }_{p}<\infty$ .)

Note that any normed vector space is a metric space with induced metric $d\left(x,y\right)=∥x,-,y∥$ . (This follows since $∥x,-,y∥=∥x,-,z,+,z,-,y∥\in ∥x,-,z∥+∥y,-,z∥$ .) While a normed vector space “feels like” a metric space, it is important to remember that it actually satisfies a great deal of additional structure.

Technical Note: In a normed vector space we must have (from N2) that $x=y$ if $∥x,-,y∥=0$ . This can lead to a curious phenomenon when dealing with continuous-time functions. For example, in ${L}_{2}\left(\left[a,b\right]\right)$ , we can consider a pair of functions like $x\left(t\right)$ and $y\left(t\right)$ illustrated below. These functions differ only at a single point, and thus ${\parallel x\left(t\right)-y\left(t\right)\parallel }_{2}=0$ (since a single point cannot contribute anything to the value of the integral.) Thus, in order for our norm to be consistent with the axioms of a norm, we must say that $x=y$ whenever $x\left(t\right)$ and $y\left(t\right)$ differ only on a set of measure zero. To reiterate $x=y⇎x\left(t\right)=y\left(t\right)\forall t\in \left[a,b\right]$ , i.e., when we treat functions as vectors, we will not interpret $x=y$ as pointwise equality, but rather as equality almost everywhere .

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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what school?
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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what is fullerene does it is used to make bukky balls
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what is the actual application of fullerenes nowadays?
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is Bucky paper clear?
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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What is lattice structure?
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or in general
Ebrahim
in general
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Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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what is biological synthesis of nanoparticles
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