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.

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How do I prove that for $\|T(x)\|=\|x\|$ for all $x$ in a vector space iff $\left<T(x),T(y)\right>=\left<x,y\right>$?

Cleaner version: $$\left \| T(x) \right \|=\left \| x \right \|,\forall x\in V\Longleftrightarrow \left \langle T(x),T(y) \right \rangle=\left \langle x,y \right \rangle$$ I know that $\left \langle ...
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I am trying to show an inequality involving the product of three inner product terms

Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as: $$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$ where $\rho$ is a real number. I ...
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22 views

Inner product in a direct sum of a dimensional space

Supposer that $V = W_{1} \oplus W_{2}$, $f_{1}$ and $f_{2}$ are inner product at $W_{1}$ and $W_{2}$, respectively. Show that there is only one inner product $f$ in $V$ such that i) $W_{2} = ...
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15 views

Orthogonal projections exercise

Let $V$ be a $n-$dimensional space with inner product and consider $W$ a subspace of $V$. If $E$ it's a projections with $Im E = W$ such that $|E\alpha| \leq |\alpha|$ $\forall \alpha \in V$ then $E$ ...
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48 views

Cauchy like inequality $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$

Problem: Prove that for real $x, y, \alpha, \beta$, $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$. I am looking for an elegant (non-bashy) ...
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22 views

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$. I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking: ...
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11 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
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25 views

Why is the standard inner product on F^n equal to this?

In the textbook that I'm using the standard inner product is defined as $$\langle x,y\rangle = \sum_{i=1}^{n}a_{i}\overline{b_{i}}$$ where $x=(a_{1}, a_{2}, {...}, a_{n})$ and $y=(b_{1}, b_{2}, ...
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27 views

Determine all values of the constants $a$ and $b$ so that the function $f$ defines an inner product.

Determine all values of the constants a and b so that the function $$f(\mathbf{x},\mathbf{y})=2x_1y_1+ax_1y_2+bx_2y_1-x_1y_3-x_3y_1+2x_2y_2+x_3y_3$$ defines an inner product. My solution. We ...
0
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1answer
23 views

What does it mean for an inner product to be conjugate linear in the second entry?

Let $G$ be a group and $L^2(G) = \{f: G \rightarrow \mathbb{C} \}$. Now define an inner product on $L^2(G)$ by $$\langle f, g \rangle = \sum_{x \in G}f(x)\overline{g(x)}$$ Where $\overline{g(x)}$ is ...
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24 views

How to decide if $Q(\underline u,\underline v) $ on $\mathbb R^2$ is inner product or not?

How to decide if $Q(\underline u,\underline v) $ on $\mathbb R^2$ is inner product or not if $$ (\underline u, \underline v) = \underline u^T A\underline v$$ where $$A = \begin{pmatrix} -1 & ...
2
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1answer
47 views

Explanation of “weight function” of inner product in Hilbert space

I am a physicist so I am sorry if the following is not written in a rigorous(or even completely right) way. As Quantum Mechanics is formed in Hilbert space, I would like to know what the weight ...
2
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2answers
29 views

Inner product, positive-definiteness

I want to define for which $a,b,c \in \mathbb{R}$ $$(X,Y) = X^T \begin{pmatrix} b+6 & 0 & 11 \\ 0 & 6 & a \\ 11 & -2 & 1 \end{pmatrix} Y + cy_1^2$$ is an inner product of ...
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17 views

Is my proof correct? Convex optimization

There's a theorem that says that if $C \subset \mathbb R^n$ is a convex set, then $x^* \in C$ is the closest point in $C$ to $y \notin C$ if and only if $(y-x^*)\cdot(x-x^*)\leq 0$ for all $x \in C$. ...
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15 views

Relating to Parseval's identity

Refering to this , my doubt is , is the other way true? I mean how can we prove( if at all it is true ) that if $<x,y>$ can be decomposed into $\sum\limits_{i=1}^n <x,x_i><x_i,y> ...
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25 views

Extending the trace inner product to all matrix (real) inner products

In ${\bf R}^{n\times p}$ we have the trace inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$ which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. All inner ...
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28 views

Fourier series and convolution

Let $f$ and $g$ be $2\pi$-periodic, piece-wise smooth functions having Fourier series $f(x)=\sum_n\alpha_ne^{inx}$ and $g(x)=\sum_n\beta_ne^{inx}$, and define the convolution of $f$ and $g$ to be ...
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3answers
35 views

How do I prove $|\left \langle x,y \right \rangle|=\left \| x \right \|\cdot \left \| y \right \|\Leftrightarrow y=cx,c\in F$

Proving $\Leftarrow$ is easy enough, it's just a matter of plugging it right in. For $\Rightarrow$, I tried changing the right side to $\left (\left \langle x,x \right \rangle \cdot\left \langle y,y ...
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2answers
43 views

For a linear map $T: \Bbb R^2 \to \Bbb R^2$ what are the inner products $[\,\cdot\,,\,\cdot\,]$ such that $\alpha, T\alpha = 0$ for all $\alpha$?

For a given linear transformation $T: \Bbb R^2 \to \Bbb R^2$, how does one obtain all inner products $[\,,\,]$ on $\mathbb{R}^{2}$ such that $$[\alpha, T\alpha] = 0 ?$$ I know that for ...
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26 views

Inner product properties

how prove the properties of inner produtcs on the formula $$ \left( \sum_{j} a_{j}x^{j}, \sum_{i} b_{i}x^{i} \right) = \sum_{i,j} \frac{a_{j}b_{i}}{i+j+1} $$ to show that $(,)$ is a inner product ...
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2answers
21 views

Conjugate symmetry to prove inner product

We have to show that $$\langle p,q\rangle=\int_a^b \overline{p(t)}q(t)$$ is an inner Product. I (think) I know what to do: I have to prove linearity, conjugate symmetry and positive definiteness. I ...
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1answer
16 views

Inner Product on $P_3(\mathbb{R})$ true/false

The question was to classify each statement as true or false with justification: Define $\langle f, g\rangle$ = $\int_0^1 f'(x)g(x) + f(x)g'(x) \,dx $. Then, $\langle , \rangle$ is an inner product ...
2
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1answer
45 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 ...
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56 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 ...
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51 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 ...
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2answers
65 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 ...
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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.
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1answer
21 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 ...
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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 ...
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1answer
25 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 ...
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1answer
17 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), ...
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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 ...
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1answer
35 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; $x_1\cdot x_2 = 0$.
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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 ...
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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 ...
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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 ...
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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 $0\notin \{x^*A^{-1}x:x^*x=1\}$ ?
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1answer
30 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 ...
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1answer
8 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 $A\bar{A}$ ...
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118 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 = ...
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1answer
29 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 $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
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37 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
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3answers
56 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 ...
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1answer
39 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
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1answer
43 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 ...
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5answers
147 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
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1answer
37 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
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1answer
17 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
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2answers
43 views

Inner product space, prove $\det(A) \geq 0$ given a particular 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
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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 ...