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

2
votes
1answer
46 views

Let $U,V \subseteq \mathbb{R}^{n}$ be two subspaces with $U \cap V = {0}$, $U \oplus V = \mathbb{R}^{n}$. Then $\langle u,v\rangle$ = 0

I'm new to this board and although you guys here have helped me a lot in the past, this is my first time to ask a question here. I couldn't find anything similar so far and I'd be grateful for any ...
1
vote
2answers
65 views

problem on hilbert spaces

Let $X=C[0,1]$ with the inner product $\langle x,y\rangle=\int_0^1 x(t)\overline y(t)\,dt$ $\forall$ $x(t),y(t)\in C[0,1]$ $X_0 =\{x(t) \in X :\int_0^1 t^2x(t)\,dt=0\}$and $X_0^\bot$ be the ...
2
votes
0answers
55 views

How did Von Neumann think of the formula for scalar product?

My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by ...
2
votes
2answers
67 views

Does $\mathbf x\cdot \mathbf y = 0$ imply that $\lVert x+y\rVert_1 = \lVert x\rVert_1 + \lVert y\rVert_1$?

If x and y are orthogonal vectors and we define $\lVert x\rVert_1 =\sum^{n}_{j=1} |x_j|$, is it possible to express $\lVert x+y\rVert_1$ in terms of $\lVert x\rVert_1$ and $\lVert y\rVert_1$ ? So I ...
1
vote
2answers
48 views

Why is $|x · y| ≤ ||x||_1||y||_∞$?

So let $||x||_∞ := $ max $_{j=1,...,n}|x_j|$ and by Cauchy-Schwarz, $|x · y| ≤ ||x||_2||y||_2$ . Why then does $|x · y| ≤ ||x||_1||y||_∞$ ? I'm not sure how to show this.
1
vote
1answer
23 views

High dimension intuition, inner spaces

Let $\mathbb E$ be a euclidean space. $dim (\mathbb E) = 5$. For every $a \in \mathbb E$ define a linear transformation $T:\mathbb E \to \mathbb R$ by the standart inner product ...
2
votes
0answers
34 views

Smallest closed subspace of $A$ in pre-Hilbert spaces [duplicate]

Let be $A\subset H$ a subset of $H$ Hilbert space. I know that $A^{\perp\perp}$ is the smallest closed subspace of $H$, such that $A\subset A^{\perp\perp}$. But if $H$ is a inner product space (or ...
2
votes
1answer
28 views

Countable Complete Orthonormal Set implies countable dense subet

Let $\mathcal H$ be a Hilbert Space, let $B = \{u_j\}_{j=1}^{\infty}$ be a countable orthonormal basis. So we know that if a set is a complete orthonormal basis, the set of all finite linear ...
1
vote
1answer
21 views

Calculating a function using orthogonal projection of vector on a subspace

Let $V$ be a closed subspace of $L^2[0,1]$ and $f,g \in L^2[0,1]$ be given by $f(x)=x,g(x)=x^2.$ If $V^{\perp}=span\{f\}$ and $Pg$ be the the orthogonal projection of $g$ on $V,$ then $(g-Pg)(x), ...
0
votes
0answers
33 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 ...
0
votes
1answer
36 views

Proof of dot product = 0 (orthogonality?)

Let $x_1,x_2$ be in $\mathbb{R}^n$ How can I prove that if $$\|x_1 + x_2\|^2 = \|x_1\|^2 + \|x_2\|^2$$ then the dot product of the vectors; $x_1\cdot x_2 = 0$.
3
votes
1answer
35 views

Example of normal linear maps

Let $V$ be a real inner product space and $g:V\rightarrow V$ a normal linear map. Can you give me an example of: 1- A map $g$ that is not self-adjoint? 2- A map $g$ that is not an isometry? 3- A map ...
0
votes
1answer
33 views

Is T self adjoint and unitary?

Consider the Hilbert space $H=l^2 $over $\mathbb C$ .If $x\in l^2$,then $\displaystyle{ \sum_{i=1}^\infty}|x_k|^2<\infty$.If $x,y\in l^2$, the inner product is defined by $$\langle ...
0
votes
3answers
57 views

Proof that there exists a polynomial $q$ such that for all polynomials $p$ we have $\int_{-1}^1p(x)q(x)dx=p(2)$ [closed]

Let $V$ be the real vector space of polynomials of degree $\leq2$ and the inner product is $\langle p,q\rangle =\int_{-1}^1p(x)q(x)dx$. How do I show that there exists a $q\in V$ such that for all ...
1
vote
1answer
34 views

$A \in GL(n,\mathbb C)$ be such that $0 \notin \{x^*Ax:x^*x=1\}$ ; then is it true that $0\notin \{x^*A^{-1}x:x^*x=1\}$?

Let $A \in GL(n,\mathbb C)$ be such that $0 \notin \{x^*Ax:x^*x=1\}$ ; then is it true that $0\notin \{x^*A^{-1}x:x^*x=1\}$ ?
2
votes
1answer
33 views

How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$

If the set $\{f_1, ..., f_n\}$ is an orthonormal subset of inner product space $E$ and $f\in E$ then how can I show that: $$\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2.$$ How ...
1
vote
1answer
10 views

Bounds on the sum of the elements of unit-length complex vector

Given an $n$-element complex vector $\mathbf{x}=[x_1,\ldots,x_n]\in\mathbb{C}^n$, where $\|\mathbf{x}\|_2^2=\sum_{i=1}^n|x_i|^2=1$, I am wondering if anything can be said about the product $A\bar{A}$ ...
3
votes
2answers
121 views

Intuition for orthogonality in $\{0, 1\}^n$

In the beginning of [Kanerva 1988] a boolean algebra over $$ \{0, 1\}^n $$ with bitwise OR and AND is introduced. Example for bitwise OR: $$101 + 001 = 101$$ Example for bitwise AND: $$101 * 001 = ...
0
votes
1answer
33 views

Bilinear forms defines inner product on Hilbert Space

I have difficulties understanding the reason why when I have a self adjoint linear operator $T : \mathcal{H} \rightarrow \mathcal{H}$, and know that $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
4
votes
0answers
42 views

A is positive definite implies Ker(A) = 0?

If $A$ is a positive definite matrix can it be concluded that the kernel of $A$ is $\{0\}$? pf: R.T.P $\ker(A) = 0$, Suppose not i.e there exists some $x$ in $\ker(A)$ s.t $x \neq 0$, then $$Ax = 0 ...
3
votes
3answers
61 views

How to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not Hilbert space?

I want to show that the Banach space $\left(C[a,b],\lVert.\rVert_{\scriptsize C[a,b]}\right)$ is not a Hilbert space. So I should show that it is not an inner product space. Most likely, The ...
1
vote
1answer
40 views

Is this proof of $|x| \leq \sum |x_i|$ correct, incorrect, or flawed?

This is problem 1-1 from Spivak Calculus on Manifolds My proof of $|x| \leq \sum |x_i|$ is slightly different from others I've seen. I would like to know if the proof is correct, but also -if it is ...
4
votes
1answer
47 views

Proof that $\langle x, y\rangle = x \cdot A \cdot y^{*}$ is an inner product.

In Spence's Linear Algebra, 4th Edition book, there's an exercise in chapter 6 who asks to proof that $\langle x, y\rangle = x \cdot A \cdot y^{*}$ is an inner product in $\mathbf{C}^{2}$, with: $$ A ...
6
votes
5answers
189 views

What is the rule for using $| \cdot |$ and $\| \cdot \|$ in Cauchy-Schwarz inequality

In this widely cited and wildly popular proof of the Cauchy-Schwarz inequality, the authors write (http://www.math.lsa.umich.edu/~speyer/417/CauchySchwartz.pdf) Let $u$ and $v$ be two vectors in ...
4
votes
1answer
44 views

Linear transformations preserve the squared sum of norms of orthonormal bases

Let $V$ be some inner product vector space over $F$, let $B=\{b_1,\dots ,b_n\}$ and $C=\{c_1,\dots ,c_n\}$ be two orthonormal bases, and let $T:V\to V$ be some linear transformation. Prove or ...
0
votes
1answer
22 views

Show that $\overline{[S]}^{\bot}=S^{\bot}$, for any subset of a inner product space.

I have done a solution for this problem: Let $X$ be an inner product space and $S$ a subset of $X$. Show that $\overline{[S]}^{\bot}=S^{\bot}$. But I am note sure that my solution is correct. If ...
2
votes
2answers
61 views

Inner product space, prove $\det(A) \geq 0$ given a Gram matrix $A$

Let $K=\mathbb R$ and let $V$ be a $\mathbb K-$finite dimensional inner product space, $\dim(V)=n$. Consider $v_1,...,v_m \in V$ with $1 \leq m \leq n$. Let $A \in \mathbb K^{m\times m}$ defined as ...
2
votes
1answer
26 views

Prove an inequality involving a norm

We define the following inner product on intergrable, $2\pi$ periodic functions from $\mathbb{R}$ to $\mathbb{C}$: $$\langle f,g\rangle = \frac{1}{2\pi} \int_{-\pi}^\pi f(t)\overline{g(t)}\ dt$$ I ...
1
vote
1answer
8 views

Find an example of an infinite dimensional vector space for which the following conditions of matrix convergence are not equivalent.

I need to find an example of linear operators $A_{n}$ and $A$ on an infinite dimensional vector space with norm $\| \cdot\|$ such that the following conditions are not equivalent: (i) $\|A_n -A\| \to ...
3
votes
1answer
33 views

Integration involving Inner Product

Suppose $f: {\bf R}^n \to {\bf R}^n $be a continuous function such that $\int_{{\bf R}^n} \vert f(x) \vert \, dx < \infty$. Let $ A \in GL_n({\bf R})$. Show that $$ \int_{{\bf R}^n} f(Ax) ...
1
vote
1answer
28 views

inner product and hermitian matrices

One of my professors mentioned that since a matrix A is positive semi definite and B is hermitian, hence the inner product $<A,B>$ is real. Is this an if and only if condition? So if we know ...
1
vote
1answer
29 views

Show that the following operator (on a Hilbert space) is continuous.

"Let $\mathcal H$ be a complex Hilbert space and let $y\in\mathcal H.$ Show that the linear transformation $f:\mathcal H\to\mathbb C$ defined by, $f(x)=\langle x,y\rangle$ is continuous." ...
2
votes
1answer
67 views

Problem about convergence of linear operators leading to ergodic theorem proof

I have been assigned to do the question I've attached. I have managed to do a,b, and c. Now I have 2 questions: (I'll use normal brackets for inner product brackets) Firstly, in part (a), I used that ...
2
votes
0answers
25 views

Bilinear form over C [duplicate]

Let $f:V\times V \rightarrow \Bbb C$ be a bilinear form in a finite inner product space. Will there always be a single linear transformation $T:V\rightarrow V$ for which $f(v,u) = \langle Tv,u\rangle$ ...
2
votes
3answers
45 views

Orthogonal Projection - property of an orthogonal operator or something that needs to be proven?

I am currently working on showing that $\Vert Px \Vert_{2} \leq \Vert x \Vert_{2}$, where $x \in$ an inner product space $X$, and $P$ is the orthogonal projection operator. Also, I am supposing that ...
3
votes
4answers
458 views

Prove that the orthogonal projection operator is idempotent

Let $\{u_{1},u_{2},\cdots,u_{n}\}$ e an orthonormal basis for a subspace $U$ in an inner product space $X$. Define the orthogonal projection of $X$ onto $U$, $P:X \to U$, to be $Px = ...
0
votes
0answers
29 views

Inner products over fields other than $\mathbb{R}$ or $\mathbb{C}$

Do inner products over other fields even make sense? E.g mod 7 or something. Also if I have a complex vector space $V$ it is a also a vector space over the reals but has twice the dimension e.g ...
0
votes
1answer
46 views

Biliniear form to inner product

Let $f:V\times V\rightarrow F$ be a bilinear form in a finite inner product space V. If $F=R$, how can I prove that there exists a single linear transformation $T:V \rightarrow V$ so that for each ...
0
votes
1answer
47 views

Tricky problem regarding Bilinear Forms,

Here's another old Linear Algebra comprehensive exam question: Let A and B be two real $nxn$ matrices. Part 1) Show that if A and B are symmetric, then for any $\vec x \in R^n$, $$((A^2 + B^2)\vec ...
1
vote
1answer
50 views

Showing that $\{\sin(nx)\}_{n=1}^\infty$ is a complete orthogonal system in $C([0,2\pi])$ and $L_2[0,\pi]$.

Showing that $\{\sin(nx)\}_{n=1}^\infty$ is a complete orthogonal system in $C([0,2\pi])$ and $L_2[0,\pi]$. So set the inner product for $C([0,2\pi])$ to be $\langle u,v\rangle = \int_0^{2\pi} ...
0
votes
2answers
46 views

If the inner product $a \cdot b = a \cdot c$. Are $b$ and $c$ linear dependent?

Let $a$ be a vector, but not the zero-vector. If the inner product $a \cdot b = a \cdot c$. Are $b$ and $c$ linear dependent if the vectors $a, b$ and $c$ are 2-dimensional? I would know how to ...
0
votes
1answer
42 views

Is $|(v,\frac{Pv}{||Pv||})|=||Pv||$ when $P$ is an orthogonal projection?

Suppose $P$ is an $k \times k$ matrix that represents an orthogonal projection. Let $v$ be an $k \times 1$ vector. Let the operator $(\cdot,\cdot)$ represents the scalar product. Does this ...
1
vote
1answer
37 views

Determine whether $\langle f, g \rangle = \int_{-1}^1 f'(x) g'(x) \,dx$ is an inner product on $C^1[-1, 1]$

Let $S = C^1[-1,1]$ functions, and define $$\langle f , g \rangle = \int_{-1}^{1} f'(x)g'(x) \,dx .$$ Decide whether $\langle \,\cdot\, , \,\cdot\,\rangle$ is an inner product on $S$. To ...
1
vote
1answer
38 views

Characterization of a projection operator in $\mathbb R^n$

Let $X$ be a closed convex set in $\mathbb R^n$ and $y\in \mathbb R^n$. Suppose a projection $T_X: \mathbb R^n \to \mathbb R^n$ satisfies $$T_X( \alpha y+ (1-\alpha) T_X(y))=T_X(y)$$ for all ...
0
votes
1answer
35 views

Linear Algebra inner products & orthonormal basis

Let $$\begin{align}f(x)&=4 \\ g(x)&=−5x+1 \\ h(x)&=−2x^2+2x−6\end{align}$$ Consider the inner product $$\langle p(x),q(x)\rangle :=p(−1)q(−1)+p(0)q(0)+p(1)q(1)$$ in the vector space ...
2
votes
3answers
38 views

Eigenvalues of a particular map $\Bbb C^3 \to \Bbb C^3$ involving an inner product

Let $f:\mathbb{C}^3\rightarrow\mathbb{C}^3$ be the linear map $$f(x)=x-i\langle x,v \rangle v ,$$ where $v\in\mathbb{C}^3$ satisfies $\langle v,v \rangle =1$. What are the eigenvalues and ...
1
vote
2answers
45 views

Show $\|A\|=\sup_{x\neq 0} \langle Ax,x\rangle/\langle x,x\rangle$ for a positive operator $A$.

I have a positive operator $A$ on the Hilbert space $\mathcal{H}$. I must prove that $\|A\|=\sup_{x \ne 0}\frac{(Ax,x)}{(x,x)}$. I am only able to get one inequality: Assume $x$ is nonzero: ...
2
votes
2answers
41 views

Selfadjoint Endomorphism

Question: Let $p >1$ be an integer, let $G = \mathbb{Z}/(p)$, and let $V = \mathbb{C}^G$, which is an inner product space over $\mathbb{C}$ with inner product defined by $\langle f, g\rangle ...
2
votes
1answer
47 views

Calculate orthonormal basis using Gram-Schmidt

Our professor gave this exercise to help us review the topic we covered in class, but it seems my knowledge is not sufficient (or we didn't cover it in enough detail during class). Assume we are ...
2
votes
1answer
26 views

Proving inequality for norm of linear transformation

Stumbled upon this one in a textbook: Let there be a linear transformation $T:V\rightarrow V$ over a finite inner product space $V$. It is known that $TT^* = 7T - 12I$. How can it be proved that ...