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 $\|x\| = \| \overline{x} \|$ 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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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
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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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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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28 views

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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1answer
29 views

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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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
24 views

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
29 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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5answers
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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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35 views

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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56 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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29 views

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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4answers
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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
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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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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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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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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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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
25 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
29 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 ...
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1answer
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Can it be defined as inner product of two vectors?

I have formulated something like For $V_n$ where ${\bf x}=(x_1,x_2,x_3, \ldots,x_n)$, ${\bf y}=(y_1,y_2,y_3,\ldots,y_n)$ and I define ${\bf x} \cdot {\bf y}$ to be $(x_1+x_2+x_3+ ...
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is inner product only valid for euclid space?

is inner product only valid for euclidean space $\mathbb R^n$? I mean, basic idea of inner product is that product of two elements in vector is real or complex. So, if it is valid for other vector ...
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What does it mean that the product of two vectors produces real number? [duplicate]

I am going over inner product space. I know that linear space has an inner product as long as it satisfies $4$ conditions. And, the book says that for $x,y$ in $V$, there is a real number ...
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1answer
37 views

Can anybody explain about “inner product”?

I am studying inner product space. One thing thing that I am trying to understand is, "How you define inner product?" For example For $\mathbb R^3$, if $ x=(x_1, x_2, x_3),\, y=(y_1,y_2,y_3)$, what ...
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what does it mean "A function that takes two vectors and return scalar?

what does it mean? A function that takes two vectors and return real number is real inner product space and complex product space if it returns a complex number Can anyone give me an example for ...
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1answer
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dot product vs inner product?

For $V_n$ where $x=(x_1, x_2, \ldots, x_n)$ and $y=(y_1,y_2,\ldots,y_n)$, the dot product is defined by $x_1y_1+x_2y_2+ \cdots+x_ny_n$. In Apostol's calculs vol 2 It says that if $x=(x_1,x_2)$ and ...
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1answer
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Can anybody explain about real linear space and complex linear space?

This is definition "A real linear $V$ is said to have an inner product if for each pair of elements $x$ and $y$ in $V$, there corresponds a unique real number $(x.y)$ satisfying what we know as ...
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Finding values for which a bilinear form is an inner product

I am trying to find the values (if any) of p and q for which the following satisfies the definition of an inner product: $$ \left \langle \mathbf{z}, \mathbf{w} \right \rangle = z_1\overline{w_1} + ...
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1answer
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Continuity of operators defined via inner products.

Let $H$ be an (in general infinite dimensional) separable Hilbert space with scalar product $<\cdot,\cdot>$. Given another inner product $<\cdot,\cdot>_2$ defined everywhere on $H \times ...
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Inner product in Besicovitch space

Besicovitch space is a space constructed in the following way: We take the closure (with respect to the uniform convergence topology) of a linear span: ...
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Notation question about scalar products and bilinear forms

Quick notation question. Is it necessary to distinguish between a scalar product and say a bilinear form $A: V \times V^* \rightarrow \mathbb{R}^n$. Would it be recommended that say you define ...
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Inner product spaces, normed spaces, metric spaces and topological spaces

I am collecting theorems or properties that hold in IPS, NS, MS or topological spaces, but not all of them. The reason is that I want to create some sort of overview over the respective spaces and ...
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3answers
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Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
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1answer
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Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
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On coverings of the complex sphere

Here, everything takes place in $\mathbb{C}^d$ for some $d$, and the sphere $\mathcal{S} = \{\mathbf{x}\in\mathbb{C}^d:\|\mathbf{x}\| = 1\}$. Given $\delta > 0$, consider a collection of vectors ...
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Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
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1answer
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Variation of orthogonal vectors

It is given that inner product $$ \left\langle a(t),b(t)\right\rangle =0,\quad \forall t\in[0,T] $$ where $a(t), b(t)\in \mathbb{R}^n$. If $\dot{a}(t)$ is known, is there a way to find an expression ...
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When is an analytic function in $L^2(\Bbb R)$?

Suppose $f:\Bbb R\to\Bbb C$ is real analytic. In order for $f$ to be in $L^2(\Bbb R)$, clearly all terms in the power series cannot be positive since $f$ would diverge at $\pm\infty$. Likewise, the ...
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2answers
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Clarification on Some Definition of Inner Product Space

Suppose $V$ is finite-dimensional Real vector space and $T\in \mathcal{L}(V)$. Suppose that $V$ has a basis $(e_1,e_2,\ldots, e_n)$ of eigenvectors of $T$, every element of $V$ can be written as a ...
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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 ...
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Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n ...
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Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
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Is every basis of a finite-dimensional vector space orthonormal with respect to some inner product?

Given a real or complex vector space $V$ and a (finite) basis $B$ of it, does it always exist an inner product on $V$ such that $B$ is an orthonormal basis with respect to it? The question is ...
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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 ...