# Questions tagged [inner-product-space]

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the “dot” product often used in vector calculus.

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### Define inner product on dual space.

V is a Hilbert space By Riesz Representation Theorem: $\forall f\in V^*\exists v$ s.t $f=l_v$ where $l_v(x)=<x,v>$ and $||l_v||=||v||$(Using this fact can check that norm of dual space ...
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### if $A \in R^{n \times n}$ , $A > 0$ and $b \in R^n$ then the function $\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$ is convex in $R^n$

Show by direct estimates that if $A \in R^{n \times n}$ , $A > 0$ and $b \in R^n$ then the function $$\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$$ with $x$ is convex on $R^n$. My ...
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### Is the norm of a vector given by dividing it by the square root of the inner product with itself?

I've known the norm (aka. unit vector in the direction of) a vector (let's call it $v$) to be given by the formula: $v/\sqrt{v \cdot v}$, where $\cdot$ is the dot product. But can this be ...
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Let $V$ be an inner product space, $T: V \to V$ a linear map, and $A=M(T,P,P)$ for an orthonormal basis $P=[v_1,...,v_n]$ (where $M(T,P,Q)$ is the matrix representation of $T$ with respect to $P$ and $... 1answer 41 views ### Differential Equation by Minimization Suppose we want to solve$u + xu' = 0$, which has the general solution$u = \frac{C}{x}$, by minimizing the length squared of$u + xu'$. This should work due to the positive definite condition of ... 3answers 95 views +50 ### For$\langle Tx, x \rangle \geq \|x\|^2$, prove a solution exists for$Tx = y$. Edit I've posted this a couple other times, so I now plan on deleting those, and just using this one. Here's the original problem:$H$is a real Hilbert space. Let$T: H\longrightarrow H$be a ... 1answer 43 views ### Are there trilinear inner products? Is there such a thing as a "trilinear inner product"? The definition of an inner product is: Let$H$be a vector space over$\mathbb{K}\in \{\mathbb{R,C}\}$. An inner product is a map$\langle \...
The inner product of two vectors $\vec a$ and $\vec b$ of n dimensions, is given by, $$(\vec a \,, \vec{b}) =a_1b_1 + a_2b_2 + a_3b_3 + \,\,...\,\,+a_nb_n$$ If a function is considered to be a ...