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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Construct an orthogonal family of three polynomials

Let $(p, q) = \int_{-1}^1 p(x)q(x) dx$ be an inner product on the vector space of all polynomials. Construct an orthogonal family consisting of three polynomials.
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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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51 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
18 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 ...
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Inner product: $(x,z)=(y,z)\implies x=y$?

We've talked about inner products in our last tutorial and couldn't really get answered the following questions: Let $(\cdot,\cdot)$ be any inner product. If $(x,z)=(y,z)$ for all $z$ of any given ...
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When is the sign of inner products preserved?

I'm interested in the following question: Let $E$ be a real Euclidian space. What are the linear transformations $f$ of $E$ that preserve the sign of inner products? That is, for all vectors ...
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26 views

Is the norm on $\ell^\infty$ induced by an inner product?

Let $\ell^\infty$ be the normed space of all bounded sequences $x \colon= (\xi_n)$ of all bounded sequences of complex numbers, with the norm defined by $$\Vert x \Vert_\infty \colon= \sup_{n \in ...
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Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
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Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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How to compute $proj_wu$; $u$ vector onto $W$ span

Let $u = (1,-2,1,6)$ in $R^4$, and let $W$ = span${(1,1,-1,0),(1,1,0,0)}$ . Compute $proj_wu$ . My Question: Since this is not an orthogonal basis, should I use the Gram-Schmidt process to convert ...
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How do you prove that T and U are the same linear transformation on an inner product space V?

Is it enough to show that $<T(x),y>$ = $<U(x),y>$ for any x and y in V?
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Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always hold?

Let $G$ be a group and $H < G$. Let $\Psi$ be a character. Let $\mathbb{I}$ be the trivial representation Does $\langle \Psi, \mathbb{I} \rangle_G=\langle Res_H\Psi, \mathbb{I} \rangle_H$ always ...
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1answer
46 views

Proof using self-adjoint property

Let $V \;$ be an inner product space over a field $\Bbb{F}$ and let $T:V\to V$ be a self-adjoint linear map. Prove that $V = \operatorname{ker}(T)\oplus\operatorname{im}(T)$. All I can think of is ...
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13 views

Showing something involving integrals is an inner product

I have this problem: Let $C([0,1])$ be the real vector space of continuous functions on the interval [0,1]. Show that $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
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56 views

Suppose $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on $V$ such that…

Suppose $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ are inner products on $V$ such that $\langle v,w\rangle_1=0$ if and only if $\langle v,w\rangle_2=0$. Prove that there is a ...
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Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by…

Suppose $p>0$. Prove that there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $\|(x,y)\|=(x^p+y^p)^\frac{1}{p}$ for all $(x,y)\in\mathbb{R}^2$ if and only if ...
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Orthogonality and inner product [closed]

Let $A\in M_2(\mathbb{R})$ a positive definite matrix and the application $F:\mathbb{R}^2 \times \mathbb{R}^2\rightarrow \mathbb{R}$ $$F(x,y)=y^tAx$$ If ...
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Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
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Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS [closed]

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
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27 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
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An example of non euclidean inner product [closed]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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Find an orthogonal basis of inner product

Let's define dot procduct $<A,B>=Trace(A B^T)$ over $M_{n \times n}(\mathbb{R})$ Find basis or system of equations describing an orthogonal $W^\perp$ subspace to subspace $W$ which consist of ...
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Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
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Getting perpendicular distance by Gram Schmidt Process

Use the Gram-Schmidt process to find the perpendicular distance from the points to the corresponding lines in the problems. a. point $(0,0)$ to the line through $(1,1)$ and $(3,0)$ b. point $(-1,0)$ ...
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48 views

Abstract Linear Algebra Inner Product [closed]

Let $u\in\mathbb{R}^n$ be a vector such that $\|u\|=1$ (for the usual inner product). Prove that there exists an $n\times n$ orthogonal matrix whose first row is $u$.
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Show that $\| u - v \|^2 = \| u - P_U(v) \|^2 + \| v - P_U(v) \|^2 $ and minimize $d(u, v)$

i) Let $\left(V, \langle\ ,\ \rangle\right)$ be an inner-product space, $v \in V$, and let $U$ be a subspace of $V$ with the orthogonal projection map $P_U$. Show that $ \| u - v \|^2 = \| u - P_U(v) ...
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Killing forms and Hermitian inner products

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...
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Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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Prob. 9, Sec. 3.2 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Here is Prob. 9 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $V$ be the vector space of all continuous complex-valued functions on ...
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$x \perp y$ if and only if $\Vert x + \alpha y \Vert \ge \Vert x \Vert$ for all scalars $\alpha$

Here's Prob. 8 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that in an inner product space, $x \perp y$ if and only if $\Vert x + ...
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Is the norm operator between normed spaces ever induced from an inner product?

Assume $(V,\| \|_V),(W,\| \|_W)$ are both finite dimensional normed spaces. We have the induced operator norm on $Hom(V,W)$. When does it occur that this norm is actually induced from some inner ...
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Parametrization of the split orthogonal group O(n,n)

I would like to find or construct an explicit parametrization of the $2m$-by-$2m$ matrix representation of the real indefinite orthogonal group $O(m,m)$ associated to the bilinear form with matrix ...
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Inner Product Space and Linear Mapping Theorem

I'm having some trouble proving the following theorem: Let $($$X$,$\langle\cdot | \cdot\rangle$$)$ be an inner product space and $f: X \to \mathbb{R}$ a linear mapping. Prove that there exists a ...
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What can we say about the inner product of two Cauchy sequences?

Let $(x_n)$, $(y_n)$ be two Cauchy sequences in an inner a real or complex product space $X$, and let the sequence $(\alpha_n)$ be given by $$ \alpha_n \colon= \ \langle x_n, y_n \rangle \ \ \ \mbox{ ...
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40 views

Is this a valid dot product?

So I have a homework problem, where $$ F(x,y) = \sin(y) (4 x ~ \mathbf{i} + \mathbf{j}) ~ \text{and} ~ \mathrm{d}{S} = - x ~ \mathbf{i} + y ~ \mathbf{j} + \mathbf{k}, $$ and I need to find the dot ...
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Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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Proofs involving orthonormal basis

Suppose that $V$ is an inner product space. (a) Show that if $\{e_1, . . . , e_n\}$ is an orthonormal basis for $V$ , then $$||v||^2=\sum_{i=1}^{n}|\langle v|e_i\rangle|^2\quad \quad \text{for every ...
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What is an example of a non-negative Hermitian form which is still not an inner product?

I was reading the definitions: Let $X$ be a vector space and $f: X \times X \longrightarrow \mathbb K$, where $\mathbb K = \mathbb R$ or $\mathbb C$. $f$ is said to be a Hermitian form on $X$ if: ...
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35 views

Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$

Suppose $M$ is a closed subspace of a Hilbert space $X$. Let $x\in X$. Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$ My Try: First of all I am confused ...
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Proving that if $M \in M_n(\mathbb{R})$ and $T(X) = MX$ is an isometry, then $M$ is orthogonal

Let $\mathbf{M}$ be a matrix in $V = M_n(\mathbb{R})$ and $T:V \rightarrow V$ be a linear operator so that $T(\mathbf{X}) = \mathbf{MX}$, $\forall \, \mathbf{X} \in V$. Considering the following ...
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Characterise the set of inner products which are preserved by a given automorphism?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
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38 views

Prove that the following integrals are equivalent.

In my linear algebra course, we are looking into inner product spaces. The following came up with regards to an inner product on a subspace of the infinitely-differentiable real functions. Let ...
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1answer
48 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have ...
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39 views

Unitary Matrices and the Hermitian Adjoint

I saw in a definition for unitary matrices, that for a complex matrix being unitary if $M: \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ is unitary, or: $\langle Mv, Mw \rangle = \langle v,w \rangle$ ...