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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Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
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Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...
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In Inner Product Space ( not complete), dose a closed linear subspace equal to the the orthogonal complement of its orthogonal complement? [duplicate]

It is apparently that this holds in Hilbert space, but I can not prove this for general inner product space or find a counterexample. (The only not complete inner product space known to me is $L^2$ ...
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How do I integrate $\langle\nabla u,\nabla v \rangle$ in arbitrary dimensions?

I am trying to show that if $u_n$ are eigenfunctions of the Laplacian operator that make up an orthonormal basis of $L^2$, then $u_n\sqrt{\lambda_n}^{-1}$ form an orthonormal basis of $H^1_0$. I ...
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is the inner product of two functions an absolute value?

My notes state that if we have two functions, $f$ and $g$, defined over the same domain, $\Omega $, then the inner product is the quantity: $$ \langle{f,g}\rangle =\int_{\Omega}f(x)g(x)dx $$ One of ...
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Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
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Showing equivalence of the positivity condition of inner products

I have no idea how to prove this, first off, because I don't think I understand the question. Isn't the second case not true for v = 0. Show that for real vectors spaces $V$ with $V$ $\not= {0}$, ...
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Notation question: What does $\langle X, - \rangle$ exactly mean?

Let $ \langle \cdot, \cdot \rangle$ be an inner product on $\mathbb{R}^n$ Then according to my course notes $$X \mapsto X^{b} = \langle X, - \rangle$$ is an isomorphism from vector fields to ...
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$x$ is orthogonal to $y$ iff $\|x+ay\| \geq \|x\|$ where $x \in C$

Show that in an inner product space, $x$ is orthogonal to $y$ iff $\|x+ay\| \geq \|x\|$ where $x \in C$. Proof: LHS: If $x$ is orthogonal to $y$, then $\langle x,y\rangle =0$. Let $a \in \mathbb C$. ...
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Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
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Why $\langle y,x\rangle+\langle x,y\rangle=2\mathrm{Re}\langle x,y\rangle$? And the rules of using absolute value, inner production and norm?

Let V be an inner product space over F, x,y∈V. In the proof of triangle inequality, my textbook uses $$\|x+y\|^2 = \langle x,x \rangle + \langle y,x \rangle + \langle x,y \rangle + \langle y,y \rangle ...
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Inner products Axioms

Apologies for my use of angular brackets, I don't know how to do it properly. Suppose $p(t) = a_{0}+a_{1}t^2+a_{2}t^2, q(t) = b_{0}+b_{1}t+b_{2}t^2 \in P_{2}(\mathbb{R}).$ Define $\langle p \;| \;q ...
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Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
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Inner product space on $\mathbb{R}^n$, prove norm is differentiable

Let $\langle \cdot , \cdot \rangle$ be an inner product on $\mathbb{R}^n$ and define $N(X)=\langle x,x \rangle ^{1/2}$. Prove that $N$ is differentiable at every point except $x=0$ and $$(DN)(x)= ...
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Proving an inequality on $\sum_{1\leq i,j \leq n} \langle c_i ,c_j \rangle \times \langle l_i ,l_j \rangle$

This is a question that stumped me during an exam I took today. Let $c_1,...,c_n,l_1,...,l_n$ be vectors of $\mathbb R^n$ and $\langle .,.\rangle$ denote the dot product. Prove that ...
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Interesting special functions identity involving the inner product of real spherical harmonics with a cosecant weight function

In spherical coordinates $\Omega=(\theta,\phi)\in[0,\pi]\otimes[0,2\pi]$, define the inner product $$C_{L_1m_1}^{L_2m_2}:=\left\langle Y_{L_1m_1},\rho,Y_{L_2m_2}\right\rangle=\int ...
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31 views

Inner product on direct sum of Hilbert spaces

Let $H_1$ and $H_2$ are two different Hilbert spaces then how can we define the inner product on $H_1\oplus H_2$
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Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
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Integral with scalar product question

Quick question about inner products and integration: Assume that we have a function $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ where $\Omega \subset ...
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Some kind of projection in a non-orthogonal basis

Sorry if the title sounds convoluted, I couldn't find any better. In $R^d$, let $(e_1,\ldots, e_d)$ be a basis. Show there exists $(a_1,\ldots, a_d)$ d vectors of $R^d$ such that $$\forall x \in ...
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Prove of inner product space and orthonormal system's necessary condition to be complete

I have no idea how to start, anything would help, thank you!
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Multivariable calculus - explain what the teacher did

The teacher gave this exercise: Find $D_f(a)$ when $f: \mathbb R^n \to \mathbb R$, $f(x)=<x,\xi>^2$ where $\xi \in \mathbb R^n$. What I did: I wrote it as $$f(x)= (\sum_{i=1}^{n}x_i ...
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Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
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Duality pairing and difference with inner product in Hilbert spaces

My question is an extension to the post Acting of a dual pairing in Sobolev Spaces. Here duality pairings were discussed and even given explicit examples. Let $U$ and $V$ be Hilbert spaces such that ...
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$T$ is a linear operator on a IPS $V$ which has a basis $\beta$. Prove that $A_{ij} = \langle T(v_j),v_i \rangle$

I have trouble understanding a proof on textbook and I would appreciate your help! Corollary. Let $V$ be a finite-dimensional inner product space with an orthonormal basis $\beta = \{v_1, v_2, ...
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Linear Algebra quick Question over inner product space

In an inner product space, not necessarily $\mathbb R^n$, there are vectors $a$ and $b$ such that $||a||\cdot ||b|| < |\langle a,b\rangle| $ Is this never true?
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Linear Algebra Explanations on true and false.

1.Could someone prove that if a set of vectors in a $p$-dimensional vector space $Q$ is a spanning set for $Q$, it is a basis. 2.If $T$ is a linear transformation from $\mathbb R^3$ onto $P_2$, then ...
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Quick Question on showing a function is an inner product

I just have a quick question How come =p(1)q(1)+p(2)q(2) is an inner product but =p(1)q(1)+p(2)q(2)-p(3)q(3) is not?
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Linear Algebra Quick Question on orthonormal basis and inner product

I have a question asking to find an orthonormal basis of $p_2$ with respect to the inner product =2 X integral from 0 to 1 p(x)q(x)dx. What do I do with the 2 in front of the integral? When I solve ...
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Finding an orthonormal basis for the space $P_2$ with respect to a given inner product

I am so confused on what to do for this question. The questions asks to find an orthonormal basis of $P_2$, the space of quadratic polynomials, with respect to the inner product $$ \langle p, ...
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If $\|u\| \leq \|u+av\|$ for all $a \in F$, How can I show that $\langle u,v\rangle=0$?

If $\|u\| \leq \|u+av\|$ for all $a \in F$, How can I show that $\langle u,v\rangle=0$? I know a standard solution uses $\operatorname{Re}$ and $t =$ something but was wondering if there was ...
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show that $\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' (t)\rangle$

Let $\gamma,\eta:[a,b]\to \mathbb R^n$ be continuous, differentiable, curves. show that $$\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' ...
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How do I find the Jacobi matrix?

I've never done questions like these, so I would very much like some help. We are given a function $f: \mathbb R^n \to \mathbb R$ given by $f(x)=\langle x,\xi\rangle^2$ where $\langle\,,\rangle$ is ...
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inner product space definition

I have some problem in the definition of inner product space. The book I use to learn in linear algebra and its application 4th edition (David C.Lay) In the chapter 6.7 it define the inner product ...
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Finding orthogonal complement

Let $X$ be an inner product space and let $x\in X$. $M=\{z\in X:\langle z,x\rangle=0\}$. I want to find $M^{\perp}$ and $M^{\perp \perp}$. Clearly, $\{x\}\subset M^\perp$. Thus $M^{\perp ...
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Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
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Inner-product question

Let $V$ be $\mathbf{R}^2$ equipped with usual inner product, and $v$ be a nonzero vector. $S_v(u)= u- 2 \frac{\langle u,v\rangle}{\langle v,v\rangle } v$ and $\Phi$ be a non-empty set of unit vectors ...
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Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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Prove that orthogonality in Euclidean space is geometrically perpendicularity?

Is this simply true by definition (that is, taken as axioms?) How would one to prove that for $||\vec{x}||=1$ and $||\vec{y}||=1$, if $(\vec{x},\vec{y})=0$, then $\vec{x}\perp\vec{y}$? In other ...
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$||u||\leq ||u+av|| \Longrightarrow \langle u,v\rangle=0$

Prove that $\langle u,v\rangle=0\Longleftrightarrow ||u||\leq ||u+av||$. So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints ...
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Continuity of a multivariate function

I'm trying to show that $\langle , \rangle$ is continuous on $V{\times}V$, ($V$ an inner product space). I've tried approaching it by showing $\langle\vec x,\vec y\rangle\rightarrow\langle\vec a,\vec ...
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Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
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Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
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find the distance between 2 functions

f(x) = $e^x$. g(x) = $.5(e-1/e) + 3x/e$. How do you find||f-g||. The inner product is defined as $\int_{-1}^1 f(x)g(x) dx$. I've tried this: $\int_{-1}^1 (e^x - (.5(e- e^{-1} + 3x/e)))^2dx$. This ...
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Prove $\langle x,x \rangle < 0$ or $\langle x,x \rangle > 0$ for all $x \neq 0$

[Added by PLC: This question is a followup to this already answered question.] Keep the axioms for a real inner product (symmetry, linearity, and homogeneity). But make the fourth be $$\langle x,x ...
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Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v ...
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Distribution Property of Dot Product

Since the dot product has the property that for three vectors $a,b,c$ $a \cdot (b+c) = a \cdot b + a \cdot c$ Is that also true for $(a+b) \cdot (a+b) = a\cdot a + 2a\cdot b + b\cdot b$ ? Thank ...
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For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
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Show a linear transform is self adjoint - check my answer

We are given $T:V \to V$ a normal linear transform (meaning $TT^*=T^*T$) We are also given $T^2=T$. Show that $T$ is self adjoint (meaning $T^*=T$). What I did I think I may have done something ...
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$T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, that ...