0
votes
2answers
29 views

Express Norm Using Inner Product

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.
3
votes
2answers
68 views

Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
1
vote
2answers
40 views

For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
0
votes
3answers
66 views

Is any norm induced by some inner product? [duplicate]

It is a well-know fact that an inner product induces some norm. How about the converse? I think it's false but I can't think of an example. I'm thinking some properties like the parallelogram law ...
2
votes
1answer
56 views

On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
2
votes
2answers
61 views

Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
1
vote
0answers
77 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
1
vote
3answers
105 views

If every inner product space can be converted into a norm space, then why is there a distinction between the two?

If every inner product space can be converted into a norm space, then why is there a distinction between the two? $$\|x\| = \sqrt{\langle x,x\rangle }$$
5
votes
1answer
71 views

Prove, that f is a linear map.

$U,V$ - Euclidean spaces $f:U \rightarrow V$ $f(0)=0$ $ \forall _{u,v \in U}:d(f(u),f(v))=d(u,v)$ Prove that $f$ is a linear map. I'm thinking about something like this: $||f(u+v)|| =d(f(u+v),0) = ...
0
votes
1answer
71 views

Show that orthogonal complement is trivial

I have this subspace of $C[-1,1]$ with inner product $\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$ E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\} $$ need to prove that $E^\bot=\{0\}$
2
votes
1answer
37 views

construction of normed linear spaces from inner product spaces

How do i construct a normed linear space from a inner product space and verify that what i have suggested is true, that is, a test of verification. Moreover, does the norm have to satisfy the ...
1
vote
2answers
101 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
3
votes
2answers
556 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 ...
3
votes
2answers
73 views

Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.

1) Can it happen that $\|x+a\|=\|x-a\|=\|x\|+\|a\|$ when $a\ne0$? 2) How large can $\min(\|x+a\|,\|x-a\|)/\|x\|$ be when $\|x\|\ge \|a\|$? (For a inner-product space, the answers are no and ...
1
vote
1answer
203 views

Relation between Normed space and inner product

Below is what i have proved: If $V$ is a normed vector space over $\mathbb{R}$ satisfies parallelogram equality, then there exists an inner product $\langle \bullet,\bullet\rangle$ such that ...
3
votes
3answers
445 views

difference between normed linear space and inner product space

I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other ...
1
vote
0answers
52 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
14
votes
1answer
499 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 $P,Q \in \mathbb{R}[X]$. ...
1
vote
1answer
346 views

Distance between real finite dimensional linear subspaces

Is there a usual distance between linear subspaces ($V,W$) of an n-dimensional normed vector space with inner product? In the case of hyper-planes one could use the angle (based on the inner product ...
6
votes
1answer
204 views

If the unit sphere of a normed space is homogeneous is the space an inner product space?

Consider a normed vector space $V$. Suppose that for every pair of unit vectors $v,w$ there exists a linear isometry which sends $v$ to $w$ (and leaves the subspace spanned by $v$ and $w$ invariant). ...
1
vote
2answers
126 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 ...
6
votes
1answer
891 views

An example of a norm which can't be generated by an inner product

I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can ...
2
votes
1answer
74 views

Norms on inner product space over $\mathbb{R}$

Definition of the problem Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. Prove that for all $x,y\in E$ we have $$ \left(\left\Vert ...
2
votes
1answer
447 views

Cauchy-Schwarz Inequality proof (for semi-inner-product A-module).

I am reading a proof of the following Cauchy-Schwarz Inequality and I don't understand one part of the proof: Theorem: Let $A$ be a $C^*$-algebra and let $E$ be a semi-inner-product $A$-module. Then ...
8
votes
1answer
106 views

Inner product space over $\mathbb{R}$

Definition of the problem I have to prove the following statement: Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. prove that for all $x,y\in ...
2
votes
0answers
132 views

Different topologies on a normed/inner product space

Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm. Define: $A$:={topologies that can make the norm continuous}, $B$:={topologies that can make the ...
5
votes
2answers
3k views

Is a norm a continuous function?

Is a norm on a set a continuous function with respect to the topology induced by the norm? Is a topology on the set that can make the norm continuous (i.e. the topology that is compatible with the ...