0
votes
1answer
22 views

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...
2
votes
2answers
37 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
1
vote
1answer
24 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
0
votes
0answers
52 views

about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
1
vote
0answers
37 views

inequality about Inner product and norm

If $m\times n~(m<n)$ matrix $A$ satisfy the following condition $(1-\delta)||s||_2^2\leqslant \|As\|^2_2\leqslant (1+\delta)\|s\|_2^2$ for all the $n \times 1$ vector with no more than $k$ nonzero ...
2
votes
3answers
338 views

Is every normed vector space, an inner product space

Let $V$ be a vector space over $\mathbb{C}$. If $V$ is an inner product space, then $V$ is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}\,\,$). Now if $V$ is normed, does it follow that ...
0
votes
1answer
61 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
3
votes
2answers
94 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
0
votes
1answer
146 views

norm induced by inner product and triangle inequality

Let $\langle\cdot,\cdot\rangle$ be a scalar product on a space $X$, and let $\lVert \cdot\rVert$ denote the norm induced by this scalar product. I need to show that for $x,y\in X$, $\lVert ...
2
votes
2answers
398 views

Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle ...
0
votes
0answers
37 views

Formulas involving the square of a norm

Why does $$\|x-y+\alpha z\|^2=\|x-y\|^2+2\alpha\langle x-y,z\rangle+\alpha^2 \|z\|^2$$ but $$\|x-z+\theta z-\theta y\|^2=\|x-z\|^2+2\theta\langle x-z,z-y\rangle +\theta^2 \|z-y\|^2?$$ Why is there ...
0
votes
0answers
52 views

Is it general to say “norm” to mean 2-norm when it is on an inner product space?

Let $V$ be an inner product space over $\mathbb{F}$. If one defines $\lVert \bullet \rVert$ as $\sqrt{\langle \bullet, \bullet \rangle}$, then $\lVert \bullet \rVert$ is a norm on $V$. However, if ...
2
votes
1answer
86 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
2
votes
2answers
30 views

Clarification with a identity

I have the following inequality which comes in proof of triangle inequality. $\|a+b\|^2=\|a\|^2$+$\|b\|^2+2\Re\langle a|b\rangle\le\|a\|^2+\|b\|^2+2|\langle a|b\rangle |$ I don't know where the ...
1
vote
2answers
42 views

Showing a function is a norm.

I'm trying to prove that $\Vert v\Vert :=\langle v,v\rangle^{1/2} $ defines a norm, but I'm having trouble with the triangle inequality. $\Vert u+v\Vert=\langle u+v,u+v\rangle^{1/2}=(\langle u,u ...
5
votes
2answers
292 views

Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
14
votes
1answer
481 views

Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
0
votes
1answer
504 views

prove that a function is an inner product

I would appreciate some assistance in answering the following problems. We are moving so quickly through our advanced linear algebra material, I can't wrap my head around the key concepts. Thank ...
2
votes
0answers
65 views

Showing inner product comes from a norm defined using Polarization Identity [duplicate]

Possible Duplicate: Norms Induced by Inner Products and the Parallelogram Law Trying to prove if an norm satisfies the Parallelogram Law then a norm arises from an inner product. Let ...
3
votes
2answers
135 views

Inequality involving norm and inner product

I am stuck proving this trivial inequality: on a real inner product space, $(||x||+||y||)\frac{\langle x,y\rangle}{||x|| \cdot ||y||}\leq||x+y||$ I have tried to square both sides and use the Cauchy ...
1
vote
1answer
172 views

Representing with Hilbert Schmidt Norm

Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...
1
vote
2answers
119 views

Does $(f,Tg)_{L^2}$ define an inner product space?

To my understanding inner product $$(f,g)_{L^2(\mathcal{D})} = \int_\mathcal{D} f(\boldsymbol{x})g(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x},~~\mathcal{D} \subset \mathbb{R}^N$$ defines an inner ...
1
vote
0answers
107 views

How to define an inner product for normed space? [duplicate]

Possible Duplicates: Norms Induced by Inner Products From norm to scalar product Given a normed space $(X, \|\cdot\|)$. Is it possible to define an inner product $\langle \cdot, \cdot ...
47
votes
3answers
7k views

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 $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...