Tagged Questions

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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Question on Norm and Inner Product

Polarisation identity states that $\langle x, y\rangle = \frac{1}{4}\|x+y\|^2 - \frac{1}{4} \| x - y \|^2$. And this is proven by expanding the terms on the right using $\|x\|^2 = \langle x,x\rangle ...
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Reproducing kernel Hilbert space, Inner Product

Let $F$ be a reproducing kernel Hilbert space, where inner product is defined as $f_1 = \sum_{i=1}^N k(\cdot,x_i)$ and $f_2 = \sum_{i=j}^M k(\cdot,y_j)$ then $ \langle f_1,f_2 \rangle ...
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Showing thislinear operator on an inner product space is its own transpose

Let $H$ be the inner product space of continuous real valued functions defined on $[0,1]$ where $(\alpha\mid\beta)=\int_{0}^{1} \alpha(u)\beta(u)du$ Put $K(s,t)=\min\{s,t\}-st$. Define $T∈L(V,V)$ by ...
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$x \cdot x$ 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 ...
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2answers
34 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
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How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
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1answer
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The inner product of two functions

If the inner product of two real functions is defined as: $$\langle f,g \rangle = \int_{-\infty}^{\infty} f(x)\cdot g(x) \ \text{d}x$$ Given $$\langle f,g \rangle=\langle f,1\rangle$$ What does it ...
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14 views

Product of dot products of two vectors

I have a product of innerproduct/dot product of two vectors. $ \langle u_i,v_j \rangle\cdot\langle x_i,y_j\rangle$. Is there any transformation/decomposition such that I can combine $u_i$ with $x_i$ ...
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24 views

Product of inner products

Is product of innerproduct again a inner product of two vectors? For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?
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42 views

Differentiating $\langle Ax,x\rangle$

If $f\colon\Bbb R^n\rightarrow\Bbb R^m$ and $g\colon\Bbb R^n\rightarrow\Bbb R^m$ are differentiable at a point $x_0\in\Bbb R^n$, and $F(x)=\langle f(x),g(x)\rangle$, then ...
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Is this sufficient for continuity?

Assume you have a map $\phi : V \rightarrow \mathbb{C}$, where $V$ is a complex vector space. Now, if we have $\phi(\lambda x) = |\lambda | \phi(x)$ and $\phi(x+y)^2+ \phi(x-y)^2 = 2\phi(x)^2 +2 ...
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1answer
24 views

What is the generalized form of this identity and how to interpret it?

I have learnt that for any inner product space of $\mathbb{C}$, we have $$\langle f,g\rangle=\frac{1}{4}\Big[||f+g||^2-||f-g||^2+i\big(||f+ig||^2-||f-ig||^2\big) \Big]$$ I know how to prove it, but ...
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0answers
80 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 ...
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Find a basis for a subspace of an inner product

Consider the vector space P$_2$(C), with inner product defined by $\langle{p(x)}$,$q(x)\rangle$ = $\int_0^1{p(x)\overline{q(x)}}dx$ Let W = {p(x) $\in$ P$_2$(C) : p'(0) = 0}. You may assume, without ...
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1answer
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Cauchy sequence of vectors when dotted with another vector gives a Cauchy sequence of scalars?

My question is related to vector spaces with an inner product defined (the space is not necessarily complete i.e. not a Hilbert Space) So imagine I have a Cauchy sequence of vectors ...
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1answer
22 views

Inner product inequality

Is there any inequality for $x^TAy$ where $x$ and $y$ are vectors and $A$ is positive definite matrix. For example: $x^TAy\ge k||x||||y||$ where $k$ is a coefficient of (min or max) eigenvalue of ...
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1answer
13 views

Association of a vector space to metric, normed and inner product spaces

There is a nice visual representation of mathematical spaces from this post: I am not quite sure how vector spaces fit into this image. I know metric space is not necessarily a vector spaces, but ...
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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 ...
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36 views

Normed Space and Hibert Space Problem

Anyone could describe me, why this is True? Suppose $(H, \|.\|) $ is a normed space. the norm $\|.\|$ induced by an inner product if and only if Parallelogram law is valid. Regards.
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matrix representation of the basis transform by Gram-schmidt orthogonalization

Gram-schmidt orthogonalization transforms a basis to an orthogonal basis. This transform of bases is linear. How can we write down the matrix representation of the transform then? Thanks.
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1answer
17 views

Coordinates of a vector under a basis in a Hilbert space?

Given an arbitrary basis $\{m_1, \dots, m_n \}$of a Hilbert space $H$ (or just think it as $\mathbb R^n$, and I think the methods should be the same) with given inner product, how can we find the ...
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1answer
20 views

Derivative involving inner product

How would I take the derivative of a function $$f(x) = < x,x >=x^{T}x?$$ The answer seems to be 2x but I don't know how to explicitly show this other than saying "there are 2 x's being operated ...
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Is $\|x\| = \| \overline{x} \|$ in an inner product space?

Suppose $X$ is a complex inner product space of complex valued functions that is closed under conjugation. Is it true that $\|x\| = \| \overline{x} \|$ for all $x$? If not, is there a simple ...
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1answer
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A question involving inner product on a Hilbert Space

Let $H$ be a hilbert space, and let $x,y \in H$. If $\langle x,z \rangle=\langle y,z\rangle$ for all $z \in H$, then $x=y$. Is this statement true or false? and why?
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Finding the maximum of an integral of a function with given constraints.

This comes from Rudin's Real Analysis text. The first part of the problem asks us to compute $\displaystyle\min_{a,b,c}\int_{-1}^1|x^3-a-bx-cx^2|dx$ (which I have done). Now it asks us to find ...
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1answer
33 views

Finiteness of the Supremum of Inner Product of Two Finite Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
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32 views

Is it possible to find the norm fuction of a space from an inner product already defined for it?

I'm a noob on the subject of functional analysis. As the title of the question says: Is it possible to find the norm fuction of a space from an inner product already defined for it? e.gr.: Suppose ...
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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), ...
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Orthonormal Basis for $[5,1+t]^{\perp}$

Consider the vector space $\Bbb{V}=P_3(\Bbb{R})$ of the real polynomials of degree less or equal 3, with the inner product given by $$\langle f,g\rangle=\int_0^1f(t)g(t)dt,\forall f,g\in\Bbb{V}.$$ ...
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1answer
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Why is $||v||$ defined as $\sqrt{\langle v, v \rangle}$ and not just $\langle v, v \rangle$?

In inner product spaces, you can create an induced norm $||v||$ from the inner product by defining $$||v|| = \langle v, v \rangle^\frac{1}{2}$$ But often (in proofs and whatnot) it's nicer to ...
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Maximize $ax + by + cz$ given $x^2 + y^2 + z^2 = k^2$. Write answer as Schwartz inequality for dot products.

Maximize $ax + by + cz$ given $x^2 + y^2 + z^2 = k^2$. Write the answer as the Schwartz inequality for dot products $(a, b, c) \cdot (x, y, z) \le \_\_\_\_\_\_\_\_ \ k$. I'm stuck on this problem. I ...
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0answers
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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 ...
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1answer
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How do I prove that an inner product has a bijection $ w \rightarrow f$

Let $\mathcal{H} = \{f:X \rightarrow \mathbb{R} : \exists w \in \mathbb{R^d}, f(x) = \langle w,x \rangle_{\mathbb{R}^d}, \forall x \in X \} $ where $\langle \cdot , \cdot \rangle$ denotes inner ...
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1answer
28 views

Multivariable calculus, inner products

I am trying to solve this question. I have considered ith component and replaced it with $v_i/(v_i^2)^{1/2}$ and the summation form of the dot product, but cannot see how the RHS falls out, can ...
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122 views

Relationship between inner product and norm

I understand that there can be many different types of norms (e.g. mean norm, Cartesian norm, supremum norm etc). Are there also other types of inner products apart from $(x,y)= \sum_{j =1}^n x_j y_j$ ...
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Proving a subspace is a closed subspace of $C[0,1]$ with inner product?

Consider the inner product space of continuously differentiable functions $C^1 [0,1]$ with the inner product: $$<f,g> = \int^1_0 f(x) \overline{g(x)} dx + \int^1_0 f'(x) \overline{g'(x)} dx$$ ...
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basic inner product exercise

I'm a little confused on how to perform this inner product. We have a generic formula of: $p^T \frac{\partial f}{\partial w}$ In the class example, we had that w was a vector, f a scalar, and p a ...
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2answers
34 views

Appolonius' identity

Verify by direct calculation that for any elements in an inner product space, $$\|z-x\|^2+\|z-y\|^2=\frac 12\|x-y\|^2+2\|z-\frac12(x+y)\|^2$$ How can I derive this identity without using the ...
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Extension from $\Bbb{R}$ to $\Bbb{C}$

i want to prove the following Lemma: Let $\mathcal{H}$ be an $n$-dimensional complex Hilbertspace, $H_1$ its unit sphere and $p:H_1\rightarrow[0,1]$ a proability distribution. Assume that for every ...
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how inner products are defined on a vector space?

How do mathematicians define inner product on a vector space. For example: $a = (x_1,x_2)$ & $ b =(y_1,y_2) $ in $ \mathbb{R}^2.$ Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. ...
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1answer
49 views

relationship between discrete and continuous time inner product

My question regard the relationship between discrete and continuous inner product $\langle f(x), g(x)\rangle =\int_a^b f(x)\overline{g(x)}dx=\lim_{N\to \infty}\sum_{i=0}^N ...
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2answers
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How would I do this? Geometry with dot products [closed]

Points P = (1, 2, −1) and Q = (3, 2, 1) and the vector n = (1, 1, 3). Using a subtraction and a dot product show that Q is not on the plane through P and perpendicular to n.
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1answer
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Prove the following version of the polarization identity:

I need to show $<x,y> = \frac{1}{2 \pi} \int_{-\pi}^\pi ||x + e^{i \theta}y||^2 e^{i \theta} d\theta$ where $<x,y>$ is inner product. So far I have: $\frac{1}{2 \pi} \int_{-\pi}^\pi ||x + ...
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0answers
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What is a orthogonal basis for constant function?

Constant function has one basis, but I think there is a way to find out orthonormal basis for constant function by letting inner product = 1. but what about orthogonal basis? Having orthonormal ...
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55 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: ...
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2answers
29 views

Product over a vector space

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms. However, none of those axioms talk about the product of two vectors. Is ...
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22 views

Can anyone please explain the meaning of this questions?

what I am struggling is as follows. If Xn(t) =t^n for n=0,1,2,3,4, prove that inner product of (Xn, Xm)=(m+n)! Where does m come from and how shall I approach this problem ?
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1answer
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What axioms does this definition fail to satisfy?

For two polynomials, f,g If I defined inner product of these two polys to be $$(f\cdot g) = \left|\int_0^1 f(x)g(x)dx\right|$$ (does this satisfy inner product? I think it is not. $x\cdot ...
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1answer
21 views

Why is this definition of inner product violating the axioms?

for all real polynomials , I defined inner product of two polys to be (f.g)=f(1)g(1). It seems to me that it does not violate any of axioms. If there is , can you tell me for what axioms are ...
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2answers
28 views

Can you show me why there has to be a conjugate sign?

this is about inner product. For 2 dimensional complex linear space, I dont see why (x.y)=conjugate of (y.x) and (x.cy)= conjugate of c (x.y) isn't is just same as when you do for real linear ...