Linked Questions

5
votes
1answer
2k views

Finding the inner product given the norm [duplicate]

Possible Duplicate: Norms Induced by Inner Products and the Parallelogram Law So suppose we are given a norm on a vector space. If the Parallelogram law holds does that automatically mean we ...
3
votes
0answers
528 views

How does the parallelogram law imply the existence of an inner product for a given norm? [duplicate]

Possible Duplicate: Norms Induced by Inner Products I am trying to prove to that if a norm of a vector space satisfies the parallelogram law ($\| \vec x + \vec y \|^2 + \| \vec x - \vec y\|^2 = 2 ...
2
votes
0answers
385 views

Prove that the norm of $E$ is generate by the inner product $\langle x,y \rangle =\frac{1}{4}\left(||x+y|^2-||x-y||^2\right)$ [duplicate]

Let $E$ a normed linear space such that: $$\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 $$ Prove that the norm of $E$ is generate by the inner product $$\langle x,y \rangle =\frac{1}{4}\left(\|x+y\|^2-\|x-y\...
2
votes
0answers
217 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 $||\cdot||$ ...
1
vote
0answers
89 views

Parallelogram law, dot product [duplicate]

Prove that if $||\cdot||$ satisfies $||u-v||^2 + ||u+v||^2 = 2(||u||^2 + ||v||^2)$ , then $u \cdot v = \frac{1}{2} (||u+v||^2 - ||u||^2 - ||v||^2)$ is dot product and $||u||^2 = u \cdot u$. I've ...
0
votes
0answers
63 views

Prove that $f(x,y)$ defines an inner product [duplicate]

Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ . We suppose that the norm satisfies the Parallelogram law. Prove that: $$f(x,y)=(1/4)[(\left\lVert x+y \right\...
62
votes
3answers
9k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric space} &...
19
votes
6answers
22k views

Relationship between inner product and norm

I understand that there can be many different types of norms (e.g. mean norm, Cartesian norm, supremum norm etc). Are there also other types of inner products apart from $(x,y)= \sum_{j =1}^n x_j y_j$ ...
13
votes
3answers
1k views

Relation between metric spaces, normed vector spaces, and inner product space.

I am wondering what exactly is the relationship between the three aforementioned spaced. All of them seem to show up many times in: Linear Algebra, Topology, and Analysis. However, I feel like I'm ...
21
votes
1answer
5k 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 ...
19
votes
1answer
9k views

$\ell_p$ is Hilbert if and only if $p=2$

Can anybody please help me to prove this: Let $p$ be greater than or equal to $1$. Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p<\infty\}$ of all $p$-summable sequences ...
8
votes
3answers
8k 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 $...
10
votes
2answers
1k views

From norm to scalar product

In the Eculidean Space, one can automatically define a norm if a specific scalar product is given. On the contrary, it's not always true. A p-norm is a scalar product if and only if p=2. My question ...
8
votes
2answers
4k views

Parallelogram law valid in banach spaces?

It is known that the parallelogram law $\|x-y\|^2+\|x+y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds in any space with an inner product (the norm being induced by this inner product). Is this formula valid in ...
5
votes
3answers
685 views

Is all normed space also inner product space?

1) I know that all inner product space is also a normed space with the norm induce by the scalar product, but is the reciprocal true ? I mean, is all normed space also a inner product space ? 2) I ...

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