Articles of inner product space

Show linearity of this map

We have the following maps on a complex vector space $V$ $\phi : V \rightarrow \mathbb{C}$ and $g : V^2 \rightarrow \mathbb{C}$ where $\lambda \in \mathbb{C} , x,y,w \in V$. $\phi $ satisfies that $\phi(\lambda x ) = |\lambda|^2 \phi(x)$ and $\phi(x+y) + \phi(x-y) = 2(\phi(x) + \phi(y))$ and the map $g$ that is defined […]

Cauchy-Schwarz for integrals

One of the exercises (3.2) of Izenman’s Modern Multivariate Statistical Techniques is for $A \subset \mathbb{R}$, $$\left( \int_{A}fg\right)^2 \leq \left(\int_{A}f^2\right) \left(\int_{A}g^2\right)$$ where $f: A \to \mathbb{R}$ and $g: A \to \mathbb{R}$ are such that $f^2$, $g^2$ are integrable. Now I’ve already proven this for a general inner product, i.e., over a vector space $V$ where […]

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 then$||x+αy||^2=||x||^2+|α|^2||y||^2 \ge ||x||^2 $ so $||x+αy|| \ge ||x||$ . But the second direction if $||x+αy|| \ge ||x||$ for any scalar […]

Show $\alpha$ is selfadjoint.

This question already has an answer here: $TT^*=T^2$, show that $T$ is self-adjoint 2 answers

$||u||\leq ||u+av|| \Longrightarrow \langle u,v\rangle=0$

Prove that $\langle u,v\rangle=0\Longleftrightarrow ||u||\leq ||u+av||$. So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints would be greatly appreciated. Please no answers, I just want a small nudge to get me in the right direction, I’ve been stumped on this for a little bit. […]

Kähler form convention

I’ve been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I’m not missing something. Let’s look at this from a purely linear algebra perspective: let $h$ be a Hermitian inner product on a complex vector space. Should […]

Indefinite inner product spaces?

Hilbert spaces require a positive definite inner product and are very well studied. Have vector spaces with a real “inner product” been studied at all, and if so under what name? In other words $\langle x|y\rangle $ is a real number, not just positive definite, and $\langle x|x\rangle = 0$ does not imply $|x\rangle = […]

Norm and inner product

Is always square of the norm of a vector is same as the inner product of that vector with itself? In Probability theory we frequently use $L^P$ norm: $\|X\|=E^{1/p}\left(X^p\right)$. But don’t we still use $E(XY)$ as the inner product? In that case, $\langle x,x\rangle=EX^2$ which is not equal to the square of the norm when […]

Proving a given formula for projection matrix

In an $n$-dimensional inner product space $V$, I have $k$ ($k\le n$) linearly independent vectors $\{b_1,b_2,\cdots,b_k\}$ spanning a subspace $U$. The $k$ vectors need not be orthogonal. Then I was told that the projection of an arbitrary vector $c$ onto $U$ is given by $$P_Uc=A(A^TA)^{-1}A^Tc$$ where $A$ is the $n\times k$ matrix with column vectors […]

Inner product is jointly continuous

I’m attempting another exercise from my notes: Show that an inner product on an inner product space is jointly continuous with respect to the induced norm:if $v_n \to v$ and $w_n \to w$ as $n \to \infty$, then $\langle v_n, w_n\rangle \to \langle v,w \rangle$ as $n \to \infty$. I’d not heard jointly continuous before […]