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

0
votes
1answer
18 views

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.
0
votes
1answer
32 views

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 ...
1
vote
0answers
20 views

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) ...
1
vote
0answers
34 views

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 ...
-1
votes
1answer
36 views

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.
0
votes
1answer
64 views

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.
0
votes
0answers
25 views

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 ...
1
vote
1answer
34 views

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 ...
0
votes
1answer
29 views

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, ...
2
votes
1answer
55 views

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), ...
2
votes
2answers
52 views

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 $\Vert \cdot \Vert_2$ ...
1
vote
1answer
30 views

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 ...
1
vote
1answer
25 views

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 ...
2
votes
2answers
52 views

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 $V$ ...
3
votes
2answers
45 views

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 $x,y \in H$ ...
1
vote
1answer
82 views

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 ...
1
vote
1answer
14 views

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 ...
2
votes
1answer
25 views

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 ...
0
votes
0answers
21 views

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 ...
1
vote
1answer
34 views

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 ...
5
votes
2answers
82 views

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 = ...
1
vote
1answer
53 views

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 ...
2
votes
1answer
12 views

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 ...
1
vote
3answers
45 views

$\langle A,B\rangle = \operatorname{tr}(B^*A)$

"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 $B$ ...
2
votes
1answer
36 views

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 )
0
votes
1answer
64 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 ...
1
vote
1answer
30 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 ...
0
votes
0answers
24 views

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 ...
1
vote
0answers
19 views

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 ...
2
votes
3answers
53 views

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 ...
2
votes
2answers
51 views

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 ...
0
votes
1answer
29 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 ...
11
votes
2answers
259 views

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, ...
0
votes
1answer
31 views

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 ...
0
votes
1answer
42 views

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 \ ...
0
votes
0answers
20 views

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 ...
2
votes
2answers
27 views

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?
0
votes
1answer
31 views

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 ...
1
vote
1answer
52 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 ...
1
vote
1answer
17 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 $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
3
votes
1answer
62 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 ...
1
vote
2answers
55 views

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 ...
-2
votes
1answer
60 views

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 ...
0
votes
1answer
48 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 ...
-1
votes
2answers
108 views

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?
0
votes
0answers
23 views

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 ...
2
votes
0answers
28 views

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 ...
3
votes
2answers
36 views

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 $(-1,0)$ ...
1
vote
1answer
55 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 $u$.
1
vote
1answer
54 views

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) ...