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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Is the Fourier transform a conformal map on $L^{2}$?

I read that a conformal map is one that preserves the angles. I know nothing more about conformal maps. I don't know where to find a generalized definition of a conformal map, but I guess that if ...
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Complex inner product linearity

Let $V$ be an inner product space over $\mathbb{C}$. Is the expression $$ \newcommand{\<}{\langle} \newcommand{\>}{\rangle} \<v,\lambda u\> = \bar{\lambda}\<v,u\> = ...
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Theorem 3.8-1 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Do we really need the completeness of the space?

Here's Theorem 3.8-1 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Every bounded linear functional $f$ on a Hilbert space $H$ can be represented in terms of the inner ...
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Inequality involving inner product and an othonormal set of vectors

$ \newcommand{\ip}[2]{\left\langle #1,#2 \right\rangle} $ Here is the statement of the problem: Suppose that $V$ is a real inner product space with an inner product $\langle\cdot,\cdot\rangle$, and ...
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On dual spaces and inner products

Let V be a vector space over $\mathbb{C}$ equipped with an inner product $\langle\, , \rangle:V\times V\mapsto\mathbb{C}$. I need to prove that any linear function $\phi:V\mapsto\mathbb{C}$ (element ...
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In euclidean space, $\forall x\in E:\|f(x)\|\le \|x\|$ implies $\ker(f-id)\oplus \mathrm{Im}(f-id)=E$

Le $E$ be an euclidean space, $f\in\mathscr L(E)$, such as $\forall x\in E:\|f(x)\|\le \|x\|$. Show that $\ker(f-id)\oplus \mathrm{Im}(f-id)=E$. I've tried to show that $\ker(f-id)\perp ...
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Theorem 3.6-2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications:” Does the converse hold if the space is not complete?

First, a definition: Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$. Now theorem 3.6-2 in Kreyszig states the ...
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$l^p$ space not having inner product

I know that $l^2$ space is a Hilbert space. But for other $l^p$ spaces, where $p\geq1$, I have to show that they do not satisfy the parallelogram equality. But, I can't find appropriate sequences ...
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Prove every isometry on an odd- dimensional real product space has 1 or -1 as an eigenvalue.

This is a question from Axler. I was hoping for some help. It seems easy to understand, but I don't know where to go about on proving this.
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Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
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Which of the following determines an inner product over the space of real continuous functions with continuous first derivatives? [closed]

I have a textbook question which I have no clue how to solve it. " Which of the following determines an inner product over the space of real continuous functions with continuous first derivatives? ...
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To prove that on $C([0,1])$, the integral $\int_{0}^{1} f(x)g(x)dx $ defines a scalar product.

Now, for the given operation to be a scalar product, I know I need to check four conditions. Here's what I have done so far: $\langle f,g\rangle $ = $\int_{0}^{1} f(x)g(x)dx$ = $\int_{0}^{1} ...
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What's the advantage of using bra-ket notation over inner-product notation?

Both notations look pretty similar, and appear similar when undergo algebraic operations. Apart from personal taste (aesthetics) concerning commas ($\langle \phantom{\cdot},\phantom{\cdot} \rangle$) ...
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$A$ be a non-empty closed convex subset of a Hilbert space $H$ , is the distance from $A$ always attained at a unique point in $A$ ?

Let $A$ be a non-empty closed convex subset of a Hilbert space $H$ , then is it true that for every $b \in H$ , $\exists$ unique $x_b \in A$ such that $||x_b-b||=d(b,A)=\inf \{||b-x||:x \in A\}$ ?
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Covariant metric tensor of a subspace

Suppose $f_1,f_2$ and $f_3$ are vectors in a vector space $V$ with a dot product. Me assume that the vectors are linearly independent. What does it mean to find the covariant metric tensor of ...
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When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$). I was wondering when the inner product $ \langle gI, t ...
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22 views

On the proof of the continuity of the inner product.

I am having problems with the following proof and I need to fill in some details: I understand that continuity is being proven by the sequence definition but I do not get why (a) follows ...
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Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
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Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$?

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$? My thinking is: $\langle u+v+w,v \rangle =0 \iff \langle u,v \rangle + \langle w,v \rangle = -1$ How do i ...
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Intersection of positive half-spaces

Let $V$ be a Euclidean space with inner product $(\cdot,\cdot)$. Let $\{\eta_i\}_{i=1}^{n}$ be a basis for $V$ , $n$ is the dimension of $V$. I want to prove that there exists a vector $t \in V$ such ...
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Inner product space definition 3 [closed]

Everywhere it says that an inner product is the generalisation of a dot product, and that it has three (four) axioms. This is way too abstract, as it doesn't give me a precise definition. For example, ...
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Show the following Inner product problem

Im stuck in this problem it seems easy but I can't find the way to show it Show that $y \perp x_n $ and $x_n \to x$ together imply $x \perp y$ Thanks for you time.
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Eigenfunction and their orthogonality with respect to the weight function

The Eigenfunction and their orthogonality with respect to the weight function $$\sigma$$ is defined as $$\int _a{}^b\phi _n\text{(x)}\phi _m\text{(x)$\sigma $(x)dx=0}$$. Given that I have some ...
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Show that $\{x\in V| \langle x,e \rangle=0 \forall e\in E\} =\{y\in V ~| ~y\perp w_i, 1\leq i \leq k \}$

Let $E$ be subset of a vector space $V$. Let $B =\{w_1,\dots,w_k\}$ be a basis for $E$. Prove: $E^\perp =\{y\in V | y\perp w_i, 1\leq i \leq k \}$ Is my proof correct? Define two sets: (a) ...
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Lemma 3.3-7 and Theorem 3.6-2 in Kreyszig's “Introductory Functional Analysis With Applications”: What if completeness is lost? [duplicate]

Let $X$ be an inner product space, and let $M$ be a non-empty subset of $X$. Then we have the following: (a) If the space of $M$ is dense in $X$, then $M^\perp = \{0 \}$, that is, $x \in X$, $x ...
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Prove norm does not come from inner product.

I know I have to show it does not satisfy the parallelogram law but I don't know how to apply it.
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Prove norm doesn't come from inner product.

Please help me prove this. I'm not sure how to apply the parallelogram law to the norm.
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Show: If $v \in E^{\perp}$ then it can be written as $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$

(i) Assume that $B = \{w_1,\dots,w_k\}$ is an orthogonal basis for $E$. Let $v \in E^{\perp}$ such that $v\neq O_{V}$. Prove that $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$ for some nonzero $w\notin E$ and ...
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Proving that the norm of $f'(y)$ is attained at $\pm\frac{\nabla f(y)}{\|\nabla f(y)\|}$.

Consider a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$ and a point $y\in \mathbb{R}^n$ such that $\nabla f(y)\neq 0$. Prove that there exists an unit vector $x_0\in\mathbb{R}^n$ such that ...
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Inner products and maxima

Let $a_1,\; ...\;, a_n$ and $b_1,\; ...\;, b_n\in \mathbb R$ be positive real numbers. Find $$ max \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ and $$ min \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ over $x_1, ...
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An inequality about inner product in $\mathbb{R}^2$.

Let $a_i,b_i,r_i,s_i$ be positive integers for $i\in\{1,2\}$. $r_i$ and $s_i$ are non-zero for $i\in\{1,2\}$. Let $a=\left(\frac{1}{a_1},\frac{1}{a_2}\right), ...
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Intersection of Hilbert spaces

Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ ...
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Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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If a subspace M is closed and dense in an inner product space V, does V = M?

The question said, if M is closed and dense in V, what conclusions can be drawn about M and V? I am assuming that these sets must be equal just by intuition and trying to visualize it. This is because ...
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Proving a sequence is a Cauchy Sequence

(Quick note: I see there are a lot of Cauchy sequence questions but I did not see this question specifically) Suppose that the sequence $v_n, n=1,2,3,... $ of elements from an inner product space $V$ ...
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Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
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A basis for $\mathbb{C}^n$

What I want to prove: Suppose $\lambda \in (-\pi,\pi]$ are natural frequencies at time $n$. Then for every $\lambda_j$ define a vector $e_j^n = \frac{1}{\sqrt{n}} \left(e^{i ...
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A rescaled inner product inequality

I was wondering if the following inequality is true: Let $\xi_1,...,\xi_n$ be vectors in a Hilbert space $H$ and let $x_{i,j}$ be complex numbers such that $\prod x_{i,j}$ is real and $$\prod ...
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Prove relationship regarding the scalar product

For 2 vectors $a,b$ $\in \mathbb{R}^n$ and all entries in the vectors are $\geq 1$ is the following relationship true ? : $\langle a,b \rangle$ $ \leq$ $0.5 \langle a,a \rangle + 0.5 \langle b,b ...
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Linear Form and Inner Product: Proof

Any source on the proof of this one: Given the field of real or complex numbers, and an inner product on a finite-dimensional vector space over the field mentioned, if $\phi$ is a linear form on the ...
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Find directional derivative - simple

The directional derivative of $f(x,y)$ at $(1,2)$ in the direction of $\vec a =\vec i + \vec j$ is $2\sqrt{2}$. We also know that the directional derivative of $f(x,y)$ at $(1,2)$ in the direction of ...
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Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications

Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications Prob. 8, Sec. 3.5 $\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = ...
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Why is this function not an inner product space?

From Morris, A. O., Linear Algebra, an introduction (2nd edition, Van Nostrand, 1989) he gives the following as not being an inner product. $(u,v)=x_1y_1-x_2y_1-x_1y_2+2x_2y_2$, where ...
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Is there a way to recover the sum of a vector coefficients?

Assuming an inner product between two vectors $\mathbb{a}$ and $\mathbb{b}$, $\langle \mathbb{a}\cdot \mathbb{b}\rangle$=v. Is there a way by knowing v and $\sum{\mathbb{b}}_i$ to obtain ...
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$\langle A,B\rangle = \operatorname{tr}(B^*A)$

"define the inner product of two matrices $A$ and $B$ in $M_{n\times n}(F)$ by $$\langle A,B \rangle = \operatorname{tr}(B^*A), $$ where the {conjugate transpose} (or {adjoint}) $B^*$ of a matrix $B$ ...
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If I want to prove that $M^{\perp}$is a closed

If I want to prove that $M^{\perp}$is a closed Can I say because it is the inverse image of $0$ by continuos function ( projection operator )
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60 views

How can I prove the following theorem with explanation? please

How can I prove the following theorem with explanation. please For any nonempty subset $M$ of a Hilbert space $H$, the span of $M$ is dense in $H$ if and only if $M^{\perp}=\{0\}$ I read the prove ...
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1answer
25 views

Eigenvalues of a Hermitian matrix and a Herminitian form

Need some help and hints on how to prove this one: Let $F=\mathbb{R}$ or $\mathbb{C}$, and $_FV=M_{n,1}(F)$. Let $A \in M_n(F)$ be Hermitian (i.e $A^* = \bar{A}^T=A$) and $f(x,y)=x^*Ay$, for all $x,y ...
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Inner Products and Linearity

I'm currently studying inner products and understand that one of the properties of an inner product on a complex space is linearity. However, there are subtle discrepencies between my lecture notes ...
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Concentration of measure of inner product in Hilbert space?

In the finite dimensional Hilbert space of quantum mechanics (one where all vectors have norm one), is a concentration of measure phenomenon observed with the inner product of any two vectors? That ...