An inner product space is a vector space equipped with an inner product. The inner product is a generalisation of the "dot" product often used in vector calculus.

learn more… | top users | synonyms

1
vote
0answers
9 views

name, notation for “block inner product” $X^H Y$

Given a set of $k$ vectors of length $n$, $X = [x_1, \dots, x_k]$ and another set of $l$ vectors of length $n$, $Y = [y_1, \dots, y_l]$, I'd like to to compute the inner product of every combination ...
0
votes
0answers
20 views

Change of base - Hermitic matrices

This exercise comes from a university exam (http://www.ubacs.com.ar/foro/viewtopic.php?f=67&t=3079, link in spanish). I'll copy it in english for everyone. It's #3: We define in $C^{n×n}$ the ...
1
vote
0answers
15 views

It is always possible to find a norm on an inner product space E which would define the weak convergence in E?

It is always possible to find a norm on an inner product space E which would define the weak convergence in E?
0
votes
0answers
32 views

Product spaces $X = Y = \mathbb R$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. It is defined for $d_{X \times Y} : X \times Y \rightarrow \mathbb R_+$ with $$d_{X \times Y}((x_1,y_1),(x_2,y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$ ...
0
votes
1answer
10 views

Is it possible to extend the inner product from an incomplete inner product space E onto H ,such that H would become a Hilbert space?

Let E be an incomplete inner product space. Let H be the completion of E . Is it possible to extend the inner product from E onto H such that H would become a Hilbert space? I think the answer is to ...
1
vote
0answers
21 views

Show that $L^2 ([ a, b])$ is the only inner product space among the spaces $L^P([a, b])$.

I know that $L^2 ([ a, b])$ is an inner product space with parallelogram law. But I cannot prove it for $p \neq 2$.
0
votes
1answer
18 views

Linear Algebra - Inner Products, Functions, and Closet Polynomial

This is the question: Formulate the linear algebra problem of finding the closet poly $p \in span \{1, t^2\}$ to the function $f(t)=e^tcos(t)$ with respect to the L$^2$ inner product: $\lt f,g\gt ...
0
votes
0answers
23 views

Inner product identity

Let $u,v:\mathbb{R}^m\to\mathbb{R}^3$ be $C^1$ functions. I need to prove the following identity: $$ \langle \nabla u, \nabla v \rangle = \langle u,u \rangle \langle \nabla u, \nabla v \rangle - ...
1
vote
1answer
56 views

Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
1
vote
1answer
21 views

Using trigonometric identity to compute an inner product through an integral to form an orthogonal sequence of functions.

Consider the inner product space: $(C(0,L),\langle \cdot,\cdot \rangle),$ where: $\langle f,g \rangle = \displaystyle\int_0^L f(x)g(x)\, dx$. Use the trigonometric identity: $\sin(u)\sin(v) = ...
1
vote
1answer
20 views

Applying Gram-Schmidt process to a set of vectors to find first three polynomials orthogonal with respect to inner product

$.$ $\langle f, g \rangle = \displaystyle\int_{-1}^{1} f(x)g(x)dx$ Apply Gram-Shmidt process to the set of vectors $:$ {1, x, $x^2$, ...} to find the first three polynomials orthogonal with respect ...
1
vote
2answers
25 views

Inner prodcut space for complex numbers including complex conjugation

$..$ Consider inner product space : $(C, \langle \cdot,\cdot\rangle)$: where for complex numbers $..$ $\langle z_1, z_2 \rangle = \sqrt(z_1 *\overline{z_2}$) Computing $..$ $\langle 2-3i, 2-3i ...
1
vote
2answers
17 views

Computing inner products with linearity in the first argument.

Consider the vector space $..$ $(\mathbb{P},\langle \cdot,\cdot \rangle)$ where the inner product is given by: $$$$ $\langle p(x),q(x) \rangle = \displaystyle\int_{-\infty}^{\infty} p(x)q(x)e^{-x^2} ...
0
votes
0answers
32 views

Adjoint linear operators and inner products question; why does $\langle T(x),T(x)\rangle =\langle T^*T(x),x\rangle $?

I have seen this multiple times in my textbook; $\langle T(x),T(x)\rangle=\langle T^*T(x),x\rangle$; why is this true? I know the definition of adjoint is if $\langle x,T(y)\rangle=\langle ...
-8
votes
0answers
46 views

Prove that this defines an inner product on polynomial space [closed]

An inner product on the polynomial space $P_2$ is a function that associates a real number, as denoted $\langle p, q \rangle$, to each pair of polynomials $p,q$ in $P_2$ and satisfies the following ...
0
votes
0answers
85 views

If every $M\subset X$ closed is such that $M^{\bot\bot}=M$, then $X$ is Hilbert space. [closed]

If $X$ is an inner product space and if $M^{\bot\bot}=M$ for every closed subspace $M$ of $X$, then $X$ is a Hilbert Space. Can someone help-me?
21
votes
3answers
1k views

Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
1
vote
2answers
49 views

In a real inner product space $u+v+w=0\implies \langle u,v\rangle=-\frac{1}{2}$? [closed]

Let $X$ be a real inner product space and $u,v,w\in X$ be vectors of norm $1$ and $u+v+w=0$. Show $\langle u,v\rangle=-\frac{1}{2}.$
1
vote
1answer
31 views

Find an orthogonal basis for W.

Use the standard Euclidean inner product on $\mathrm R^4$. Let $W$ be the subspace of $\mathrm R^4$ spanned by $u_1 = (1, 1, 1, 1),$ $u_2 = (2, 4, 1, 5),$ $u_3 = (1, -5, 4, -8).$ Find an ...
1
vote
1answer
38 views

Kernel of adjoint and orthogonal complement images

Alright, suppose we are given $V$, a finite dimensional inner product space, and a linear map, $T:V \rightarrow V$, with its corresponding adjoint, $T^\star :V \rightarrow V$. I want to show: ...
2
votes
1answer
26 views

A conceptual question about inner product spaces for continuous functions and bases.

I hope this makes sense, I had a test yesterday and I couldn't answer this question. The question was laid out something like the following: I was provided a basis for 5th degree polynomials, ...
0
votes
1answer
25 views

Proving $\left|\left|\vec{w}\right|\right|^2 \ge \sum_{j=1}^n\left|\langle\vec{w},\vec{u}_j\rangle\right|^2$.

Proving $\left|\left|\vec{w}\right|\right|^2 \ge \sum_{j=1}^n\left|\langle\vec{w},\vec{u}_j\rangle\right|^2$. $\vec{u}_1,...,\vec{u}_j$ is a set of orthonormal vectors in an inner product space $V$. ...
0
votes
0answers
47 views

Let u and v be nonzero vectors in an inner product space V. Prove that $u-projv(u)$ is or orthogonal to v.

I do know that the question is asking to prove $\langle u-\operatorname{proj}_v(u),v \rangle =0$. But I don't know how to prove it. By the way, an inner product is not a dot product. I know how to ...
0
votes
1answer
16 views

Proofing the sum of the squre cosines of the angle between a vector and the vectors of the base.

I'm having trouble getting this proof right, its somthing like this Let $ \{e_1, e_2, ... e_n \}$ be an orthonormal basis, of a vector space with internal product $\mathbb V$, and $\theta_i$, the ...
0
votes
1answer
23 views

Scalar Products equation proof

$\langle \langle x + y, z \rangle \rangle = \langle \langle x, z \rangle \rangle + \langle \langle y, z \rangle \rangle$ It is clear when there are only $\langle \dot \ , \dot \ \rangle$ but what ...
2
votes
1answer
18 views

Solution Operator for inhomogenous Dirichlet Problem

Writing up a report, I want to understand the following solution operator better. Suppose $\Omega$ is an open bounded domain in $\mathbb{R}^d$ with boundary $\partial\Omega$ of class ...
1
vote
1answer
38 views

Inner product alternative definition

I'm trying to make an alternative (but equivalent) definition of an inner product. I prefer to use arbitrary sums with $\sum_i v_i$ instead of sum of just two vectors $v+w$, and few but clear axioms. ...
0
votes
1answer
29 views

$H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$

When $\Omega$ is a bounded open set of $\mathbb{R}^N$ with the help of Poincare inequality, we know that $H_0^1(\Omega)$ with $(u,v)_{H_0^1} = \int\nabla u \cdot \nabla v$ is a Hilbert space. ...
0
votes
1answer
12 views

Relative complement and subtraction.

Given two vector subspaces $W_1$ and $W_2$ of an inner product space $V$, and $W_2 \subseteq W_1$, the relative complement of $W_2$ in $W_1$ is defined as $W_1 \cap W_2^\perp$, and some book uses the ...
1
vote
1answer
37 views

Finding the Orthogonal Complement to a subspace

So suppose I have a vector space, $V$ which is all continuous functions on $[0,1]$. Additionally, we have an inner product over $V$ where $\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$. Now suppose I ...
0
votes
1answer
25 views

Finding the closest vector in a subspace to another vector

Let $$W=sp(\{(1,-1,0,0),(1,2,0,-1),(1,0,0,1)\}) \subseteq \mathbb{R}^4$$ Find $w \in W$ closest to $(0,2,1,0)$. Any hints on how to approach this question?
2
votes
1answer
35 views

Definition of angle between vectors in spaces with dimensions n

I am working on a problem which requires me to find certain values of the components of vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^4$ such that the angle between them is $\pi/3$ If my ...
2
votes
1answer
112 views

Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ...
0
votes
1answer
22 views

Use the orthonormality of $u,v,w$ to write the following vectors as linear combinations of $u,v$ and $w$

Let $V$ be the vector space $\mathbb R^3$ with inner product $$(v,w)=3(v_1w_1)-2(v_1w_2)-2(v_2w_1)+5(v_2w_2)-3(v_2w_3)-3(v_3w_2)+3(v_3w_3)$$ where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. Part 1 ...
1
vote
3answers
21 views

Let V be an inner product space. Prove that for any 2 vectors, (u,v)=1/4(|u+v|^2-|u-v|^2)

Let V be an inner product space. Prove that for any 2 vectors (u,v)=1/4(|u+v|^2-|u-v|^2) Thanks very much for any help
5
votes
2answers
43 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
0
votes
1answer
22 views

Necessary and sufficient conditions for an adjoint of a linear map to be the map's inverse

Let $V$ be a finite dimensional inner product space, $ \phi :V \rightarrow V$ a linear operator and $\phi^*:V \rightarrow V $ its adjoint. I wish to show: $\phi^*$ is an inverse to $\phi$ if and ...
1
vote
0answers
15 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
1
vote
0answers
31 views

Orthonormal set is a Hilbert basis $\iff$ Parseval's identity is true

Let $H$ be a Hilbert space and $\{e_k:k\in \mathbb{Z}\}$ an orthonormal set. Prove that the set is a Hilbert basis if and only if Parseval's identity is true. The direct theorem is almost ...
0
votes
0answers
62 views

When are sesquilinear forms actual inner products?

I read it was enough (and necessary) to have $\overline A^T=A$ and $A$ non-singular, for a sesqui-linear form on $\mathbb{C}^n$ to be an actual inner product. (Here $A$ is a matrix for the ...
2
votes
2answers
33 views

Bilinear Map vs Inner Product

What is the difference between a Bilinear Map and a Inner Product?
3
votes
1answer
17 views

Question on Completeness of Derived Inner Product Space

Let $(\mathcal{H},\langle{,}\rangle)$ be a separable, infinite-dimensional Hilbert space. Let $\mathcal{X}''$ denote the space of bounded sequences in $\mathcal{H}$. For a Banach limit $L$, define a ...
0
votes
0answers
22 views

Norm in $C(X,\Bbb{R})$

Let $X\subset\Bbb{R}$ a compact set and $f\in C(X,\Bbb{R})$. Define $$\|f\|_{\infty}=\sup A_f$$ with $A_f=\{|f(x)|\in \Bbb{R};x\in X\}$. Then $\|f\|_{\infty}=|f(x_0)|$, for some $x_0 \in X$, since ...
5
votes
5answers
223 views

Show: $(x+y)^4 \leq 8(x^4 + y^4)$

I wish to show the following statement: $ \forall x,y \in \mathbb{R} $ $$ (x+y)^4 \leq 8(x^4 + y^4) $$ What is the scope for genralisaion? Edit: Apparently the above inequality can be shown ...
2
votes
2answers
35 views

Writing a vector as the sum of orthogonal vectors

At the start of a proof of the Cauchy-Schwartz inequality, my lecturer wrote down the following statement: Let $V$ be an Inner Prouct Space with underlying field $\mathbb{F}$, then $$ \forall\ \ ...
1
vote
0answers
82 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...
2
votes
0answers
31 views

Inner product exterior algebra

I have to prove that if V is a real vectorial space (dimV=n) with inner product (.,.) then if we define $$ (v_{1}\wedge v_{2}\wedge...\wedge v_{k},w_{1}\wedge w_{2}\wedge...\wedge w_{k}) ...
1
vote
1answer
23 views

Composition of linear transformations that preserve angles

Given two invertible linear transformations T1,T2 in L(V) that preserve angles i.e. $\frac{(T(u), T(v))} {∥T(u)∥∥T(v)∥} = \frac{(u, v)} {∥u∥∥v∥} $. How can I show that T1T2 and T-1 also preserve ...
0
votes
0answers
15 views

Vector Derivative of Dot Product in Reflection Equation

There are several questions on this network about differentiating a dot product of vector functions of an independent scalar parameter $t$:$$ \frac{d}{d t}\left(\vec{f}(t)\cdot\vec{g}(t)\right) = ...
4
votes
2answers
107 views

Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...