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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Find an orthogonal basis of inner product

Let's define dot procduct $<A,B>=Trace(A B^T)$ over $M_{n \times n}(\mathbb{R})$ Find basis or system of equations describing an orthogonal $W^\perp$ subspace to subspace $W$ which consist of ...
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Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$.

Consider $\mathbb{C}^4$ with the standard inner-product$ < , >$. Extend $(\frac{1}{2}, \frac{i}{2} ,\frac{-1}{2},\frac{-i}{2} )$ to an orthonormal basis for $\mathbb{C}^4$. How is this possible ...
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Getting perpendicular distance by Gram Schmidt Process

Use the Gram-Schmidt process to find the perpendicular distance from the points to the corresponding lines in the problems. a. point $(0,0)$ to the line through $(1,1)$ and $(3,0)$ b. point $(-1,0)$ ...
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55 views

Abstract Linear Algebra Inner Product [closed]

Let $u\in\mathbb{R}^n$ be a vector such that $\|u\|=1$ (for the usual inner product). Prove that there exists an $n\times n$ orthogonal matrix whose first row is $u$.
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Show that $\| u - v \|^2 = \| u - P_U(v) \|^2 + \| v - P_U(v) \|^2 $ and minimize $d(u, v)$

i) Let $\left(V, \langle\ ,\ \rangle\right)$ be an inner-product space, $v \in V$, and let $U$ be a subspace of $V$ with the orthogonal projection map $P_U$. Show that $ \| u - v \|^2 = \| u - P_U(v) ...
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57 views

Killing forms and Hermitian inner products

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...
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Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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Prob. 9, Sec. 3.2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications”

Here is Prob. 9 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $V$ be the vector space of all continuous complex-valued functions on ...
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$x \perp y$ if and only if $\Vert x + \alpha y \Vert \ge \Vert x \Vert$ for all scalars $\alpha$

Here's Prob. 8 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that in an inner product space, $x \perp y$ if and only if $\Vert x + ...
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70 views

Is the norm operator between normed spaces ever induced from an inner product?

Assume $(V,\| \|_V),(W,\| \|_W)$ are both finite dimensional normed spaces. We have the induced operator norm on $Hom(V,W)$. When does it occur that this norm is actually induced from some inner ...
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32 views

Parametrization of the split orthogonal group O(n,n)

I would like to find or construct an explicit parametrization of the $2m$-by-$2m$ matrix representation of the real indefinite orthogonal group $O(m,m)$ associated to the bilinear form with matrix ...
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42 views

Inner Product Space and Linear Mapping Theorem

I'm having some trouble proving the following theorem: Let $($$X$,$\langle\cdot | \cdot\rangle$$)$ be an inner product space and $f: X \to \mathbb{R}$ a linear mapping. Prove that there exists a ...
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45 views

What can we say about the inner product of two Cauchy sequences?

Let $(x_n)$, $(y_n)$ be two Cauchy sequences in an inner a real or complex product space $X$, and let the sequence $(\alpha_n)$ be given by $$ \alpha_n \colon= \ \langle x_n, y_n \rangle \ \ \ \mbox{ ...
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45 views

Is this a valid dot product?

So I have a homework problem, where $$ F(x,y) = \sin(y) (4 x ~ \mathbf{i} + \mathbf{j}) ~ \text{and} ~ \mathrm{d}{S} = - x ~ \mathbf{i} + y ~ \mathbf{j} + \mathbf{k}, $$ and I need to find the dot ...
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84 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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30 views

Proofs involving orthonormal basis

Suppose that $V$ is an inner product space. (a) Show that if $\{e_1, . . . , e_n\}$ is an orthonormal basis for $V$ , then $$||v||^2=\sum_{i=1}^{n}|\langle v|e_i\rangle|^2\quad \quad \text{for every ...
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14 views

What is an example of a non-negative Hermitian form which is still not an inner product?

I was reading the definitions: Let $X$ be a vector space and $f: X \times X \longrightarrow \mathbb K$, where $\mathbb K = \mathbb R$ or $\mathbb C$. $f$ is said to be a Hermitian form on $X$ if: ...
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38 views

Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$

Suppose $M$ is a closed subspace of a Hilbert space $X$. Let $x\in X$. Prove that $min\{\|x-y\|:y\in M\}=max\{|\langle x,y\rangle|:y\in M^\perp , \|y\|=1\}$ My Try: First of all I am confused ...
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Proving that if $M \in M_n(\mathbb{R})$ and $T(X) = MX$ is an isometry, then $M$ is orthogonal

Let $\mathbf{M}$ be a matrix in $V = M_n(\mathbb{R})$ and $T:V \rightarrow V$ be a linear operator so that $T(\mathbf{X}) = \mathbf{MX}$, $\forall \, \mathbf{X} \in V$. Considering the following ...
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Characterise the set of inner products which are preserved by a given automorphism?

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
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42 views

Prove that the following integrals are equivalent.

In my linear algebra course, we are looking into inner product spaces. The following came up with regards to an inner product on a subspace of the infinitely-differentiable real functions. Let ...
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63 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have ...
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48 views

Unitary Matrices and the Hermitian Adjoint

I saw in a definition for unitary matrices, that for a complex matrix being unitary if $M: \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ is unitary, or: $\langle Mv, Mw \rangle = \langle v,w \rangle$ ...
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Then prove: $\vec{v} = \vec{0}$ if $\langle u,v\rangle = 0$

If $\vec{v} \in V$ such that $\langle u,v\rangle = 0$, $\forall \vec{u} \in V$. Then prove: $\vec{v} = \vec{0}$ I tired to solve by assuming that they are $\langle u,v\rangle \neq 0$ $\rightarrow$ ...
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42 views

Prove $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$

If V is the finite dimensional inner product space, then prove the following: If $u, v \in V$ are linearly dependent, then $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$ Thanks.
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65 views

Orthogonal in inner product space

Let $(X,<.>)$ is an inner product space prove that $x$ and $y$ are orthogonal if and only if $||x+αy|| \ge ||x||$ for any scalar $α$ . The first direction if $x$ and $y$ are orthogonal ...
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89 views

Cauchy-Schwarz Hermitian Inner Product Remainder

A couple weeks ago, someone showed me a proof of Cauchy-Schwarz where he ended up deriving something of the form $$|\langle a,b\rangle|^2=|\langle a,a\rangle||\langle b,b\rangle| +f(a,b)$$ Where ...
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83 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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43 views

$X$ is inner product space then its completion is Hilbert space?

I have trouble finding a way to prove that the completion of my innerproduct space $X$ is a Hilbert space. How do I know that the norm on the completion of $X$ is induced by an innerproduct? Thanks ...
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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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43 views

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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20 views

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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22 views

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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55 views

$x\perp y$ 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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36 views

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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57 views

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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141 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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21 views

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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62 views

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 ...