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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Subspace of $L^p(X,\Sigma,\lambda)$

Consider $R$-valued functions in $L^p(X,\Sigma,\lambda)$, where $X=X^1\times X^2$, $\Sigma=\Sigma^1\times \Sigma^2$ and $\lambda=\lambda^1\times \lambda^2$ For given $i$, does the subsapce $M=\{f\in ...
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Under what conditions is a linear automorphism an isometry of some inner product?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and that $T: V \to V$ is a (linear) isomorphism. When is it possible to construct an inner product on $V$ making $T$ an ...
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Complex euclidean tensor products

Say you have Euclidean vectors $\mathbf{a}=a_i \mathbf{p}_i$ and $\mathbf{b}=b_j \mathbf{q}_j$ in $\mathbb{R}^3$, with bases $\mathbf{p}_i$ and $\mathbf{q}_j$. Then you could use a typical inner ...
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Proving a Matrix Inner Product

I am given a matrix inner product on square matrices defined as $\langle A,B\rangle=tr(AB^t)$, where $M^t$ denotes the transpose. I am asked to prove that this is indeed an inner product. We go by 3 ...
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When does equality hold in this case?

Give example of two vectors $x$ and $y$ such that $$||x+y||_2^2 = ||x||_2^2+||y||_2^2$$ and $$<x,y>\neq0$$ I can't seem to find any two vectors $x$ and $y$ that satisfied both conditions at ...
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What is the rigorous justification for using inner products as a function of similarity between two vectors?

In machine learning, it is a common thing to define similarity measures, specially using the so call Kernel function. Kernel functions are defined though through inner products of feature vectors: ...
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Parallelogram law using complex inner product not adding up

Does the parallelogram law still hold in the complex case? Using the following definitions: $\langle \textbf{x}, \textbf{y} + \textbf{z} \rangle = \langle \textbf{x}, \textbf{y} \rangle + \langle ...
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34 views

What does this assumption imply in case $X$ is a complex inner product space?

If $X$ is a real inner product space and $x,y\in X$ satisfy $\|x\|=\|y\|$, then $(x-y)\perp (x+y)$. What does this assumption imply in case $X$ is a complex inner product space? My Work: I proved ...
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Prove that a) $(span(M))^\bot=M^\bot$ b) $(\overline{M})^\bot=M^\bot$

Let $X$ be an inner product space.$M\subset X$. Prove that a) $(span(M))^\bot=M^\bot$ b) $(\overline{M})^\bot=M^\bot$ My Work and problems: a) Clearly $(span(M))^\bot\subset M^\bot$. Now let ...
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$x\perp y$ iff $\|x+\lambda y\|\geq \|x\|$ for all scalars $\lambda$

Show that in an inner product space $X$ a) $x\perp y$ iff $\|x+\lambda y\|=\|x-\lambda y\|$ for all scalars $\lambda$ b) $x\perp y$ iff $\|x+\lambda y\|\geq \|x\|$ for all scalars $\lambda$ My ...
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Is $\langle A,B \rangle = \text{trace}(A^{T}B)$ undefined in $\mathbb{R}^{n\times 1}$ and $\mathbb{R}^{1\times m}$?

$A^TB$ would be a $n\times 1$ or $1\times m$ vector in each case, no? How can we sum diagonal elements if they don't exist?
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Does $\langle f+h,g\rangle=\langle f,g\rangle+\langle h,g\rangle$ hold for all elements $f, g, h$ of an inner product space?

Are there any exceptions? I was thinking proof by contradiction i.e. define $\langle f,g\rangle\ \neq0$ for two orthogonal elements of the product space, but positive definiteness would require one ...
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Inner Product Spaces and Dual Spaces

Let $G$ be the matrix of the scalar product in a basis $(\mathbb{e_{1}}, ..., \mathbb{e_{n}})$ of a Euclidean space $V$. Find the matrix of the change of base to the dual one $(f_{1}, ..., f_{n})$ and ...
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How to show that $\langle A,B \rangle = a_{11}b_{11}+a_{12}b_{12}+a_{21}b_{21}+a_{22}b_{22}$ is an inner product on $M_{2x2}$?

Let $$\langle A,B \rangle = a_{11}b_{11}+a_{12}b_{12}+a_{21}b_{21}+a_{22}b_{22}.$$ Show that this in an inner product on the vector space $M_{2x2}$? I just do not get how to prove this with ...
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Closest Vector in a Inner Product Space

Let $V$ = $\mathbb{R}^n$ Note that $\langle -,-\rangle$ defines the Inner Product on $\mathbb{R}^n$ $$\|v\| = \sqrt{\langle v,v \rangle}$$ Consider the standard Distance Function $$d(x,y) = ...
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What would be a characterization of a definite operator?

Let $V$ be an $n$-dimensional inner product space and let's call $T\in \mathcal L (V)$ definite if $$\forall x \neq0: \langle Tx,x\rangle \neq 0. $$ An obvious sufficient condition for $T$ to be ...
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Multiplication of an inner product and its conjugate

While studying Inner Product Spaces, I'm seeing that $\langle{x, y}\rangle * \overline{\langle{x, y}\rangle} =|c|^2$, where $ c=\langle{x, y}\rangle $ and $c$ is a constant. The $inner$ $product$ ...
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Why does this work? Fourier coefs. of function with min energy in window is eigenvector of window coef. matrix.

Let me begin with saying I have never got a good handle on eigenvectors and eigenvalues. My best hunch is that the eigenvectors are the 'best' basis for a linear transform along which the transform ...
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138 views

Prove that the Eigenvalues of this Matrix are in [0,1 ]

Let $E,F \subset \mathbb{R^n}$ Note that $< . >$ defines the Inner product on $\mathbb{R^n}$ Let $(e_1,....,e_k)$ and $(f_1,.....f_l$) be Orthonormal bases of E and F respectively. Consider ...
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Orthonormal Basis of Two Subspaces

Let $S,T$ $\subset \mathbb{R^n}$ Prove that it is possible to choose an Orthonormal Basis W for S and W' for T such that $W = (s_1,....,s_k)$ $W' = (t_1,.....t_m)$ $<s_i,t_j>$ = 0 if $i \neq ...
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Find $f$ such that $\int_{-\pi}^{\pi}|f(x)-\sin(2x)|^2 \, dx$ is minimal

Fairly simple question that's been bothering me for a while. Supposedly it should be simple to solve from the properties of inner product but I can't seem to solve it. Find $f(x) \in ...
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Inner Product on Linear Space of Continuous Functions

Consider the linear space of continuous functions $C[-1,+1]$ defined over the interval $[-1,+1]$. We define an inner product $\langle\cdot , \cdot\rangle$ on $C[-1,+1]$ by $$\langle ...
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Gram-Schmidt and Inner Product Spaces

Consider the matrix $A=\begin{pmatrix}2&1\\1&2\end{pmatrix}$. We define a new inner product over $\mathbb{R}^2$ given by $\langle\vec{v},\vec{w}\rangle=\vec{v}^T\cdot A\cdot \vec{w}$. Find ...
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Does $\langle Fv_j, Fv_j \rangle = \langle F^*v_j, F^*v_j \rangle$ imply that $F$ is normal?

Let $\{v_1, \ldots, v_n\}$ be an orthonormal basis of an inner product space $V$ and a linear $F:V\to V$ s.t. for all $j$: $$\langle Fv_j, Fv_j \rangle = \langle F^*v_j, F^*v_j \rangle.$$ Does it ...
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$\max <x,q>$ when $x \in H$ Hilbert and $q \in A^\bot$

How can I find $\max \langle x,q \rangle$ where $x$ is fixed in a Hilbert space $H$ and $q$ runs over $ A^\bot$ with $\|q\|=1$, $A$ a proper nontrivial subspace? (Here we assume $x \notin A, A^\bot$.) ...
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Orthonormal basis of polynomials

I am trying to find an orthonormal basis of the vector space $P^{3}(t)$ with an inner product defined by $$\langle f, g \rangle = \int_0^1f(t)g(t)dt$$ by applying the gram schmit alogorotin to ...
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Which of these two is an inner product space

I have a simple question on which am seeking help clarifying. Am looking at two inner products, one which my text says is an inner product and I find it to be not, and another my text says it is but I ...
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Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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Linearity in first argument of $\langle X,Y\rangle =X^*MY $

In paper it says that the inner product of two $n$-dimensional vectors is equal to: $$\langle X,Y\rangle :=X^*MY $$ where $M$ is Hermitian and positive definite. It is easy to proof the conjugate ...
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Find the inner product under which is the following base orthonormal

This question is inspired by the following problem If we know that the basis $B=(\mathbf{u},\mathbf{v})$, where $$\mathbf{u}= \begin{pmatrix}1 \\ 2i\end{pmatrix},\;\;\mathbf{v}=\begin{pmatrix}-i ...
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Proof for $\mathbf{v}\cdot\mathbf{v} = \| \mathbf{v} \|^2$ [closed]

Anyone have a good link for proving $\mathbf{v}\cdot\mathbf{v} = \| \mathbf{v} \|^2$? I tried ProofWiki but the site doesn't seem to be responding right now.
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Question about inner product space and positive definite matrix

I was working on problems in a textbook for my intro linear algebra class, and one of the questions asked about showing that if we have some real positive definite matrix, A, then $\lt ...
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104 views

Inner product on $C[0,1]$

Let $V = \mathcal{C}[0,1]$ with inner product $\langle v,w\rangle =\int_{0}^1 v(x)w(x) \mathrm{d}x$. Find a functional $f\in V^*$ for which there does not exist a vector $v\in V$ satisfying $f(w) = ...
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Prove that the norm in an inner product space is $ \ge 0$

Macdonald Linear and Geometric Algebra defines an Inner Product Space in the following way (pg 57): "An inner product space is a vector space with a product called an inner product. The inner product ...
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Generalization of inner product spaces (analogue to uniform spaces/locally convex spaces)

In the following I am going to devise a chart of topological spaces that contains inner product spaces, normed vector spaces, metric spaces and other related spaces. In the end there will be a gap in ...
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What does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So ...
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If $T^2=T$ then determine whether $\ker T=\operatorname{Range}\,(T)^\perp$.

Let $T$ be linear operator on a finite dimensional inner product space $V$ such that $T^2=T$. Determine whether $\ker T=\operatorname{Range}\,(T)^\perp$. I have proved that $\ker ...
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Inner product in Hilbert spaces

Considering a sequence $\{\boldsymbol{v}_k\}_{k=1}^\infty$ in a Hilbert space $\mathcal{H}$, and let $\{c_k\}_{k=1}^\infty \in \ell^2(\mathbb{N})$. Then for all $\boldsymbol{v}\in\mathcal{H}$ $$ ...
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Show that for every real-valued $L^2$ function $u$ on $S^1$ there is a real-valued $v$ in the same space such that $u + iv\in \widetilde{\mathbf H}^2$

For a homework exercise ($1.8$ in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let $u$ be a real-valued function in $L^2(S^1)$. Show that there exists a ...
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Proof of the Cauchy-Schwarz inequality in Axler's Linear Algebra Done Right

The proof of the Cauchy-Schwarz inequality in Axler's Linear Algebra Done Right (pg. 104) hinges on showing that $\|u\|^2 = \|\frac{\langle u,v\rangle}{||v||^2}v\|^2 + \|w\|^2 \tag{1}$ equals ...
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Little trouble with inner product space

I am having a little difficulty trying to solve a beginner proof in the topic of inner product spaces. The statement says, suppose $a_1,…,a_n \in \mathbb R$, Prove that $(a_1+…+a_n)^{2}/n \le ...
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Why is $\langle f+g, f+g \rangle = \langle f, f \rangle + \langle g, f \rangle + \langle f, g \rangle + \langle g,g\rangle$?

Why is $\langle f+g, f+g \rangle = \langle f, f \rangle + \langle g, f \rangle + \langle f, g \rangle + \langle g,g\rangle$? I know that from linearity: $$\langle f+g, f+g \rangle = \langle f, ...
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Possible existence of weight function $\rho (t)$

Consider $L^2[-\pi,\pi]$. We define an inner product on this space by $$\langle f,g\rangle=\int_{-\pi}^{\pi} f(t)\overline {g(t)} \, dt \quad\to(1)$$ Suppose if we introduce a weight function ...
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Enlarge $S$ to an orthonormal basis

Question: Let $S=\{ \bar{u}_1 , \bar{u}_2 \}$ be an orthonormal set in $\mathbb{R^3}$ and let $\bar{u} \in \mathbb{R^3}$. Enlarge $S$ to an orthonormal basis for $\mathbb{R^3}$ I am unsure ...
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To show that orthogonal complement of a set A is closed.

To show that orthogonal complement of a set A is closed. My try: I first show that the inner product is a continuous map. Let $X$ be an inner product space. For all $x_1,x_2,y_1,y_2 \in X$, by ...
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Finding rotate matrix which solves equation

I try to solve the following problem: given a unit vector v, find rotate matrix R such that R*v = (0,0,..0,1) (vector that it's (n-1) components are 0 and the n'th component is 1). I know that if I ...
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In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
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A real vector space is an inner product space if every two dimensional subspace is an inner product space ?

Is it true that a vector space over the field of real numbers is an inner product space if every two dimensional subspace is an inner product space ? does it have anything to do with Neuman-Jordan's ...
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Looking for a simpler proof of Day's characterization of inner-product spaces and related things

I know the theorem that if $(V,||.||)$ is a real normed linear space such that the parallelogram identity $||x+y||^2+||x-y||^2=2(||x||^2+|y||^2)$ holds , then the norm comes from an inner-product ...
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Do the statements hold in an inner product space over $\mathbb R$ as well?

Let $V$ be an $n$-dimensional inner product space over $\mathbb C$ and $f\in \mathcal L (V)$ normal. Show that: $f^2=f^3 \implies f=f^2 \implies f = f^*$ $f$ nilpotent $\implies f=0$ ...