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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Use the orthonormality of $u,v,w$ to write the following vectors as linear combinations of $u,v$ and $w$

Let $V$ be the vector space $\mathbb R^3$ with inner product $$(v,w)=3(v_1w_1)-2(v_1w_2)-2(v_2w_1)+5(v_2w_2)-3(v_2w_3)-3(v_3w_2)+3(v_3w_3)$$ where $v=(v_1,v_2,v_3)$ and $w=(w_1,w_2,w_3)$. Part 1 ...
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Let V be an inner product space. Prove that for any 2 vectors, (u,v)=1/4(|u+v|^2-|u-v|^2)

Let V be an inner product space. Prove that for any 2 vectors (u,v)=1/4(|u+v|^2-|u-v|^2) Thanks very much for any help
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Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
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Necessary and sufficient conditions for an adjoint of a linear map to be the map's inverse

Let $V$ be a finite dimensional inner product space, $ \phi :V \rightarrow V$ a linear operator and $\phi^*:V \rightarrow V $ its adjoint. I wish to show: $\phi^*$ is an inverse to $\phi$ if and ...
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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 ...
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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 ...
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50 views

When are sesquilinear forms actual inner products?

I read it was enough (and necessary) to have $\overline A^T=A$ and $A$ non-singular, for a sesqui-linear form on $\mathbb{C}^n$ to be an actual inner product. (Here $A$ is a matrix for the ...
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Question on Completeness of Derived Inner Product Space

Let $(\mathcal{H},\langle{,}\rangle)$ be a separable, infinite-dimensional Hilbert space. Let $\mathcal{X}''$ denote the space of bounded sequences in $\mathcal{H}$. For a Banach limit $L$, define a ...
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22 views

Norm in $C(X,\Bbb{R})$

Let $X\subset\Bbb{R}$ a compact set and $f\in C(X,\Bbb{R})$. Define $$\|f\|_{\infty}=\sup A_f$$ with $A_f=\{|f(x)|\in \Bbb{R};x\in X\}$. Then $\|f\|_{\infty}=|f(x_0)|$, for some $x_0 \in X$, since ...
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Show: $(x+y)^4 \leq 8(x^4 + y^4)$ Using Cauchy-Schwarz Inequality

I wish to show the following statement: $ \forall x,y \in \mathbb{R} $ $$ (x+y)^4 \leq 8(x^4 + y^4) $$ What is the scope for genralisaion? Edit: Apparently the above inequality can be shown ...
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Writing a vector as the sum of orthogonal vectors

At the start of a proof of the Cauchy-Schwartz inequality, my lecturer wrote down the following statement: Let $V$ be an Inner Prouct Space with underlying field $\mathbb{F}$, then $$ \forall\ \ ...
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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 ...
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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}) ...
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Composition of linear transformations that preserve angles

Given two invertible linear transformations T1,T2 in L(V) that preserve angles i.e. $\frac{(T(u), T(v))} {∥T(u)∥∥T(v)∥} = \frac{(u, v)} {∥u∥∥v∥} $. How can I show that T1T2 and T-1 also preserve ...
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Vector Derivative of Dot Product in Reflection Equation

There are several questions on this network about differentiating a dot product of vector functions of an independent scalar parameter $t$:$$ \frac{d}{d t}\left(\vec{f}(t)\cdot\vec{g}(t)\right) = ...
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Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
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61 views

Inequivalent norms (given by different inner products) on infinite dimensional Hilbert space.

I have this question in reviewing for my exam. Let $H$ be an infinite dimensional Hilbert space. Write down an inner product on $H$ that gives a norm inequivalent with the original norm. Is $H$ ...
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32 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 ...
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24 views

Construct an inner product space $X$ and a proper, closed subspace $Y$ of $X$ such that $Y^\perp = \{ 0 \}$.

I am looking to do the following: Construct an inner product space $X$ (with inner product $\langle \cdot, \cdot \rangle$ and a proper, closed subspace $Y$ of $X$ such that $Y^\perp = \{0\}$, ie ...
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Question on Inner Product Space

I know that an inner product space can be split into 2 vector spaces $W$ and $W^ \perp$. So if $x$ is not in $W ^\perp$. However, when I think about it, if I have a line that passes through origin in ...
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Show that the formula defines an inner product on X

Let $X=C[-1,1]$ be the space of continuous functions $f:[-1,1]\rightarrow \mathbb{R}$. For $f,g\in X$ define: $$\langle f,g\rangle_2=\int_{-1}^{1}|t|f(t)g(t)dt$$ The property i'm struggling with is ...
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31 views

Prove that $\langle v,w \rangle = (A{\bf v}) \cdot (A{\bf w})$ defines an inner product on $\mathbb{R}^m$ iff $\ker(A)=\{{\bf 0}\}$

My instructor showed the proof of this result by proving the three axioms of inner product on the given proposed inner product, then using positive definiteness to show the $\ker(A)=\{{\bf 0}\}$. I ...
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76 views

A basis $B$ is orthonormal if and only if $\langle f,g \rangle=[f]_B\cdot [g]_B$ for all $f$ and $g$ in $V$

Consider a finite-dimensional inner product space $V$. If $B$ is a basis for $V$, show that $B$ is orthonormal if and only if $\langle f,g \rangle=[f]_B\cdot [g]_B$ for all $f$ and $g$ in $V$. I'm ...
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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 ...
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12 views

totality of subset of an inner product space

How can I show that, given a subset $M$ of an inner product space $X$: If $M$ is a total set, then, $M^\perp =\{0\}$? ($M^\perp $ is the orthogonal complement of $M$)
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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 ...
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Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
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29 views

Vector Space with Scalar Product: $x\to x^{2}$ and $x\to x^{3}$

For a space with scalar product (dot product) $C[0,1]$, we have two functions$f(t):t\to t^{2}$ and $g(t): t\to t^3$. Here we have $<f,g> = 1/6$, $||f^2||= 1/5$, and $||g^2||=1/7$. My question ...
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28 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 ...
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72 views

a question about symmetric positive definite matrix and norm

If B is $n\times n$ real symmetric positive definite matrix, then $(x,y)=x^TBy$ definites an inner product on $R^n$. How to prove that $||x||=(x^TBx)^{1/2}$ is a norm on $R^n$?
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Prove that a linear transformation is negative

Let $V$ be a complex inner product space, and $T$ a skew-adjoint operator, whereby $T^*=-T$. Prove that all the eigenvalues of $T$ are purely imaginary. Prove that $T^2$ is non-positive. Prove ...
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Does the orthogonal projection theorem guarantees uniqueness of the projected space?

Given a Hilbert space $H$, and linear map $P:H \to H$ such that $P^2=P$ and for every $x\in H$ : $\|Px\| \le \|x\|$, there is a closed linear-subspace $M$ such that $P=P_M$, the projection on $M$. My ...
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Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
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Vectors inner product on $\mathbb R$

Let $u_1,...,u_k$ vectors in $\mathbb R ^ n$ such that for every i,j we have $u_i \cdot u_j < 0$, and I want to show that $k\leq n+1$. I tried somehow use the fact that each vector seperates the ...
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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 ...
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16 views

When a self-adjoin operator is invariant wrt any orthogonal projection

Given a linear operator $F$ on an inner product space $V$. If $F$ is self-adjoin and satisfies that $$ P F P^T = F $$ for any orthogonal linear transformation $P: V\to V$, is it true that $F = c I$ ...
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I need help with this proof [closed]

Let $x,y\in \Bbb R^n$. Prove that$$ \sum_{k=1}^n x_k y_k \le \|x\|_2 \|y\|_2$$
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$l^p$ norm on an inner product space

On a finite-dimensional inner product space $X$, can we define its $l^p$ norm with $p \geq 1$, as the $l^p$ norm on its coordinate space under an orthonormal basis of $X$, and the $l^p$ norm doesn't ...
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Show that the function 1/t is not in L2 (0,1]

Need some help getting started with this problem: $$f(t) = \frac{1}{t}$$ Show that $f(t)$ is not in $L_2(0,1]$, but that it is in the Hilbert space $L_{2}w(0,1)$ where the inner product is given by ...
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53 views

Can anyone explain me about real and complex euclidean space?

Is euclidean space a linear space where inner product exists? If $V$ is a complex linear space, then do there complex numbers $x,y$ in $V$ such that $\langle x,y \rangle$ is complex? And I don't ...
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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),-),\, ...
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Matrix space, with $\langle A,B\rangle=\text{tr}(AB^*)$ isn't Hilbert space, how can i find a counter example?

Generally, I'm having quite troubles thinking about counter examples. So I would love if someone could guide me into finding the example for the following question by myself (and not just giving it ...
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52 views

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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32 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 ...
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2answers
54 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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84 views

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
32 views

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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20 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 $x_i$ ...