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13
votes
2answers
913 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 ...
12
votes
1answer
445 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 ...
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 ...
9
votes
1answer
359 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 ...
6
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 ...
6
votes
1answer
303 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 ...
5
votes
2answers
422 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 ...
5
votes
2answers
401 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 ...
3
votes
4answers
724 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 ...
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" ...
3
votes
2answers
452 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 ...
2
votes
1answer
72 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 ...
1
vote
1answer
263 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 $x,y \in U,$ $$\langle x,y \rangle_U \ge \langle Tx, Ty\rangle_V.$$ ...
1
vote
2answers
466 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 ...
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 ...
1
vote
0answers
132 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 ...
0
votes
1answer
69 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 ...
0
votes
0answers
95 views

Operator Adjoints and Non-Symmetric Inner Products

Let $V$ be a finite dimensional vector space (over $C$ if that makes a difference), and let $T$ be a linear operator on $V$. Now if $(\cdot,\cdot)$ is an inner product on $V$, then it is well-known ...
0
votes
0answers
93 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?