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0
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2answers
14 views

Generalised “projection” of a metric space

Assume we have $n$ points $p_0\ldots p_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p_0) = 0$, $f(p_1) = d(p_0,p_1)$ and ...
-1
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2answers
61 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
3
votes
1answer
74 views

A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued ...
1
vote
0answers
33 views

Nonadjointable Equivariant Operators on Hermitian Vector Bundles

Let $V$ be an equivalent complex vector bundle over a homogeneous space $X$, and $D:\Gamma^\infty(V) \to \Gamma^\infty(V)$ an equivariant operator. If we put a Riemann structure on $X$, and an ...
0
votes
1answer
70 views

Is the Hodge Map Unitary?

Let $(V,< \cdot, \cdot >)$ be an inner product space over a field ${K}$. As usual, we can extend $< \cdot, \cdot >$ to a mapping on the exterior algebra of $V$ using the usual matrix ...
12
votes
1answer
493 views

A property that forces the NORM to be induced by an INNER PRODUCT

Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$, $$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$ I want to show that the norm is induced by an inner product. Any ...
2
votes
1answer
75 views

Classifying all Equivariant Bilinear Forms on a Finite-Dimensional Module

Given a finite dimensional (real) vector space $V$, and two non-degenerate bilinear forms $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$, one can use a basic linear algebra argument to show that there exists ...
6
votes
1answer
309 views

Almost orthogonal vectors in subsets of euclidean space

Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of almost-orthogonal sets ...
0
votes
0answers
99 views

Functional Analysis Generalizations: indeterminated inner product and functions over manifolds

There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
1
vote
0answers
137 views

How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?

Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...
13
votes
2answers
977 views

Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello, I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it. I can ask it in two different ways. Perhaps depending on the reader, the ...
9
votes
1answer
381 views

Unitary representations of Quantum Groups

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
3
votes
2answers
266 views

Inner product spaces, Siegel's theorem and lattices: book suggestion

Background: I am a theoretical computer scientist (PhD candidate) and have done graduate level courses in Algebra. I want to understand the following theorem from the book "Symmetric Bilinear Forms" ...
7
votes
6answers
3k views

What is a complex inner product space “really”?

This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO. To begin with, in a real inner product space we have a geometric ...
5
votes
2answers
427 views

adjoint of multiplication operator in a commutative algebra

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO. Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with ...
11
votes
2answers
2k views

Why do inner products require conjugation?

For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of ...
1
vote
2answers
478 views

Hash functions and inner product

Hi all, As a part of a research I'm working on (involving derandomization of linear threshold functions), I'm trying to understand the following problem: Is there a small (polynomial rather than ...
5
votes
2answers
403 views

Can finitely many hermitian (positive-semidefinite) operators always be embedded into a small dimensional space preserving inner product?

Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which ...
1
vote
1answer
264 views

Which linear transformations between f.d. Hilbert spaces contract the inner product?

Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$ $$\langle x,y \rangle_U \ge \langle Tx, Ty\rangle_V.$$ ...
3
votes
2answers
464 views

Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules

Shapovalov and Jantzen showed us how to construct a nice inner product on finite dimensional representations of a semi-simple Lie algebra, by simply giving the highest weight vector inner product 1 ...
3
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4answers
730 views

An inner product that makes the R-matrix unitary

So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...