Linked Questions

185
votes
4answers
46k 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 $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
60
votes
2answers
25k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
33
votes
3answers
3k views

Norm for pointwise convergence

Does there exist a norm on the space of all real-valued functions on the real line (or on an open set? a compact set?) such that convergence in this norm is equivalent to pointwise convergence?
22
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 ...
13
votes
1answer
2k views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
5
votes
3answers
780 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 ...
16
votes
1answer
2k views

Does convexity of a 'norm' imply the triangle inequality?

Given a vector space $V$ (for convenience, defined over $\mathbb{r}$), we call $d:V\rightarrow\mathbb{R}$ a norm for $V$ if $\forall \mathbf{u}, \mathbf{v} \in V$ and $\forall r \in \mathbb{R}$ we ...
4
votes
1answer
2k views

Metric Space, Normed Space, and Inner Product space hierarcy

I'm having trouble understanding the intuition of the hierarchy of metric space, normed space, and inner product space. What additional structure do I gain at every level? I'm going to list my ...
5
votes
1answer
871 views

Hierarchy of Mathematical Spaces

I really got lost among all those many different spaces in mathematics, and I got really confused what is special case of what. For example, I knew for long time vector spaces, then Hilbert spaces, ...
4
votes
5answers
284 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
5
votes
1answer
318 views

Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$

I'm reading functional analysis in the summer, and have come to this exercise, asking to show that the two spaces $l^p(\mathbb{N})^*,l^q(\mathbb{N})$ are isomorphic, that is, by showing that every $l \...
3
votes
2answers
345 views

Is a complex space more “advanced” than a “generic” real space?

For instance, does taking the square root of a complex number and its complex conjugate create a metric that "automatically" makes it an inner product space? Is a complex space more complete than a ...
3
votes
0answers
649 views

Understanding examples - metric spaces, Minkowski functionals and topologies

I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me ...
0
votes
1answer
88 views

Building Euclidean space

What's the minimum amount of extra "structure" do we need to add to the general concept of an affine space to get Euclidean space? That includes the concepts of angle and distance, in which we can ...
0
votes
2answers
62 views

Why metric vector space may have few convex sets

I am reading a answer by t.b. ,see here Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces) He(maybe she) said that "The first observation I ...