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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Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
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Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
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Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
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An example of non euclidean inner product [on hold]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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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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Abstract Linear Algebra Inner Product [on hold]

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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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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30 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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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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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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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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39 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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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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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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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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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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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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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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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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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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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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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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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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$X$ 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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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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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 $inner$ $product$ ...