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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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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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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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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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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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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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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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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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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prove of inner product space [closed]

Show that$$ L^1 (R^n ) $$under the operator $$〈.,.〉:L^1×L^1→R$$ such that $$〈f,g〉=∫_(R^n)▒〖f(x) (g(x)) ̅ 〗$$ for all$$ f,g∈L^1 (R^n )$$forms an inner product space
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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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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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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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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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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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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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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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3answers
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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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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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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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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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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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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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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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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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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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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?

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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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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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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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 axioms for ...
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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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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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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 $G$ ...