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

4
votes
0answers
225 views

Inner Product as Weighted Average

Let $\zeta=e^{2\pi i/n}$, where $n\geq3$. Let $||\cdot||$ be the norm induced by the complex inner product $\langle\cdot,\cdot\rangle$. Then $$\langle ...
3
votes
0answers
45 views

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 ...
3
votes
0answers
47 views

Shouldn't 'without loss of generality' be mentioned here?

Regarding the blocked portion my question is what if $x\ne z=y?$ Shouldn't without loss of generality be mentioned in the text?
3
votes
0answers
103 views

Inner spaces - A question about adjoints.

Question: Let $V$ be a complex vector space over $\mathbb{C}$ with inner product $\langle ,\rangle$. Let $E$ be an linear operator on $V$ such that $E^2=E$ with adjoint $E^*$. Show that $E$ is ...
3
votes
0answers
680 views

Existence of n distinct (real) roots of an orthogonal polynomial

I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof ...
2
votes
0answers
31 views

Inner product exterior algebra

I have to prove that if V is a real vectorial space (dimV=n) with inner product (.,.) then if we define $$ (v_{1}\wedge v_{2}\wedge...\wedge v_{k},w_{1}\wedge w_{2}\wedge...\wedge w_{k}) ...
2
votes
0answers
37 views

Intuition behind complex inner product

Let $f_n : \mathbb C^n \to \mathbb R^{2n}$ be defined by $$(x_1 + iy_1, x_2 + iy_2, \dots) \mapsto (x_1, y_1, x_2, y_2, \dots )$$ I am having trouble believing that $$ \langle X, Y\rangle_{\mathbb ...
2
votes
0answers
11 views

The problem of support vector machine - How to minimize $||w||^{2}$ subject to constraints of the form $\alpha w_{1}+\beta w_{2}+b\geq\pm1$

I am studying the subject of support vector machines from an online course. I am given four points and their classification $$ x_{1}=((5,4),+),\, x_{2}=((8,3),+) $$ $$ x_{3}=((3,3),-),\, ...
2
votes
0answers
47 views

Show that $(Au,Bv)=(u,A^tBv)$

Let $ A, B $ be matrices of order $ n $, and $ \vec{u}, \vec{v} $ vectors from euclidean space $ \mathbb{R}^n $, then $ (Au,Bv) = (u,A^tBv) $ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
2
votes
0answers
20 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
2
votes
0answers
32 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
2
votes
0answers
28 views

Linear group action over an hermitian space.

Let $\mathrm{h}$ be an non degenerate hermitian form on $\mathbb{C}^2$, and suppose $\mathrm{h}$ is of signature $(1,1)$. Let $A$ and $ B \in \mathrm{SL}(2,\mathbb{C})$ such that $G$ the group ...
2
votes
0answers
110 views

Weighted $L^p$ spaces and orthogonal families of polynomials

Let $I$ be a closed interval (bounded or not) of $\mathbb{R}$ and let $w\colon I \rightarrow \mathbb{R}$ be a continuous function that is positive inside $I$ and such that for every $n\in ...
2
votes
0answers
197 views

Is my formula for the change of basis of an inner product correct?

Suppose we have a real or complex finite dimensional vector space $V$ with an inner product $\langle \cdot , \cdot \rangle$ and a basis $B = \{v_1, \dots, v_n\}$. Define the matrix of the inner ...
2
votes
0answers
377 views

Gram-Schmidt Orthogonalization for subspace of $L^2$

I am a little stuck on the following problem: By using the Gram-Schmidt Orthogonalization, find an orthonormal basis for the subspace of $L^2[0,1]$ spanned by $1,x, x^2, x^3$. OK, so I have defined: ...
2
votes
0answers
145 views

Different topologies on a normed/inner product space

Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm. Define: $A$:={topologies that can make the norm continuous}, $B$:={topologies that can make the ...
1
vote
0answers
19 views

Show that $L^2 ([ a, b])$ is the only inner product space among the spaces $L^P([a, b])$.

I know that $L^2 ([ a, b])$ is an inner product space with parallelogram law. But I cannot prove it for $p \neq 2$.
1
vote
0answers
15 views

Given two dot products with the same vector in a prime finite field of 2 (Galois Field), how can one figure out future dot products?

I've stumbled upon an interesting "rule" derivation for the value of a dot product in $\mathbb{R}^{n}$ like this: Given an arbitrary vector $\vec a \in \mathbb{R}^{n}$ and the values of two dot ...
1
vote
0answers
31 views

Orthonormal set is a Hilbert basis $\iff$ Parseval's identity is true

Let $H$ be a Hilbert space and $\{e_k:k\in \mathbb{Z}\}$ an orthonormal set. Prove that the set is a Hilbert basis if and only if Parseval's identity is true. The direct theorem is almost ...
1
vote
0answers
76 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...
1
vote
0answers
27 views

Directional derivative, why is this incorrect?

A function $\vec{F}$ is dependent on the lenght (norm) of $\vec{r}$ ${\vec{F}(\vec{r})=F_{x}(\vec{r})\hat{x}+F_{y}(\vec{r})\hat{y}+F_{z}(\vec{r})\hat{z}} $, in which ...
1
vote
0answers
27 views

When can we take $A$ somehow out of $\langle x,Ay\rangle$?

Let $A$ be a linear mapping on an inner product space $V$ and $x,y \in V$. What are some cases (as general as possible) can we take $A$ somehow out of $\langle x,Ay\rangle$? (I know when $A = c I$, we ...
1
vote
0answers
29 views

Is the orthogonal complement of $V=C[0,1]$ is ${0}$ in $L^2[0,1]$

I was trying to think about the orthogonal complement of $C[0,1]$ in L$^2[0,1]$. I thought that it should be $\{0\}$ but I had only little confident with my proof so I'd like to ask you if it's ...
1
vote
0answers
30 views

Orthonormal basis Parsevals identity.

Let $O={u_1,...,u_k}$ be an orthonormal set in $V$. Prove that $O$ is an orthonormal basis if and only if Parseval's identity holds for all $v,w \in V$ i.e if and only if $$\langle ...
1
vote
0answers
37 views

$x \cdot x$ in inner product space is a quadratic form

Given an inner product space with some inner product $\cdot$ , how can I prove that $x \cdot x$ for any vector $x= (x_1,... x_n)$ is a quadratic form in $x_i$? I know how to recover an inner product ...
1
vote
0answers
91 views

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that ...
1
vote
0answers
55 views

Is it true that $V$ and $V^*$ are naturally isomorphic as finite vector spaces if $V$ is equipped with an inner product?

This is a homework question from my differentiable manifolds class: In general we know that if $\dim V<\infty$ then $V$ and $V^*$ are isomorphic because any two vector spaces with the same ...
1
vote
0answers
35 views

Gradient of an Inner Product in a more general Vector Space

I was looking at the following question: Differentiating an Inner Product that was talking about the derivative of an inner product to be: $$ \frac{d}{dt} \langle f, g \rangle = \langle f(t), ...
1
vote
0answers
19 views

How do you show the connection of reproducing kernels to feature maps?

This question is in the context of Hilbert Reproducing Hilbert Spaces and reproducing Kernels and their relation to feature maps (and machine learning). We have a Hilbert space $\mathcal{F}$ and ...
1
vote
0answers
80 views

Orthogonal Complement

"Let $\Bbb{V}$ be a vector space with an inner product $<\cdot,\cdot>$, and $S\subset\Bbb{V}$. We define the orthogonal complement of $S$, denoted by $S^{\perp}$, as follows: ...
1
vote
0answers
16 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
1
vote
0answers
59 views

Is an inner product continuous?

It is an easy question, but i want to make it clear :) Let $(V,\langle -,- \rangle)$ be an inner product space over $\mathbb{K}$. Then, is the inner product $\langle -,- \rangle:V\times V\rightarrow ...
1
vote
0answers
20 views

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ...
1
vote
0answers
50 views

Showing an inner product space is complete

I'm working through Ward Cheneys Analysis for Applications and I'm a bit stuck on this exercise from Section 2.2: Prove that if $M=M^{\perp\perp}$ for every closed linear subspace $M$ in an inner ...
1
vote
0answers
106 views

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 ...
1
vote
0answers
59 views

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?
1
vote
0answers
143 views

Hermitian positive semi-definite matrix is a Gram matrix

I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = <x_i,x_j>$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive ...
1
vote
0answers
75 views

Linear form, there exists a unique vector $z$ such that $f(w)=\langle w,z \rangle$

Let $V$ be a finite dimensional space over the field $\mathbb{F}$ with inner product $\langle \cdot, \cdot \rangle$. Then for every linear form $f: V \rightarrow \mathbb{F}$ there exists a unique $z ...
1
vote
0answers
88 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
1
vote
0answers
26 views

How can I find $f^{ad}$ along with $\operatorname{Im} f^{ad}$ and $\operatorname{Null} f^{ad}$?

Let $ f \in \mathbb R_{\leq4}[t]$ such that $f(p)=f \bigg( \sum\limits_{k=0}^4 a_k t^k\bigg):=f \bigg( \sum\limits_{k=1}^4 k a_k t^{k-1}\bigg)$ and define $\langle p,q \rangle := \int\limits_{-1}^1 ...
1
vote
0answers
39 views

Schwarz Inequality Proof: The Mystery of the Disappearing Vector

I've been scratching my head over this for about 45 minutes now, and I have no idea where why the $w$ in this proof disappeared. $\left| \left \langle u,v \right \rangle \right| \le ||u|| ||v|| $ ...
1
vote
0answers
56 views

inequality about Inner product and norm

If $m\times n~(m<n)$ matrix $A$ satisfy the following condition $(1-\delta)||s||_2^2\leqslant \|As\|^2_2\leqslant (1+\delta)\|s\|_2^2$ for all the $n \times 1$ vector with no more than $k$ nonzero ...
1
vote
0answers
56 views

Necessary and sufficient conditions for a normal operator to be self-adjoint

What are the necessary and sufficient conditions under which a normal operator becomes self-adjoint.
1
vote
0answers
42 views

$\left< u, v\right> \stackrel{?}{=} \left< v, u \right>$

For some vector space $V$, is it true that $$ \left< u, v \right> = \left< v, u \right> $$ for all $u, v \in V$? Does this only hold if $V \subseteq \mathbb{R}^n$ or if $V \subseteq ...
1
vote
0answers
53 views

A simple piece of a Lemma on Gram-Schmidt

I was looking at a proof of Gram Schmidt theorem and I saw the following lemma, it starts here: First the theorem: if $V$ is an inner product space and $X= \{x_1,\dots, x_n\}$ is a linearly ...
1
vote
0answers
64 views

Dot product with a difference maximum angle

In a program I'm making I use the dot product to get a value between 1 and 0 dependent on the angle between two vectors. The value is 0 if the vectors are parallel and 1 if they are perpendicular. ...
1
vote
0answers
37 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
1
vote
0answers
70 views

Is $\langle f \rangle $ an “inner product”?

Let $$\langle f(x,y)\rangle = \iint_S f(x,y)\,\mathrm{d}x\,\mathrm{d}y$$ I have seen the above in multiple papers as the definition of $\langle f(x,y)\rangle$. I would normally associate angle ...
1
vote
0answers
53 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
1
vote
0answers
202 views

Inner product is well-defined

Can you tell me if this is correct? Thanks. Claim: The following inner product is well-defined, i.e. finite for all $f,g \in L^2$: $$ (f,g) = \int_X f \overline{g} d \mu$$ Proof: We may assume $f,g$ ...