# Tagged Questions

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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### Self adjoint differentiation operator on vector space of polynomials and inner products.

Let $n \in \mathbb{N}$. For the following, consider the vector space of polynomials of degree less than or equal to $n$ with complex coefficients, $\mathcal{P}_n(\mathbb{C})$. a) If we equip the ...
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Is there a general rule to simplify things like $<x,y> - <x,z>$ or generally $<.,.> \pm <.,.>$ I cant find anything in my notes that talks about this.
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### Property of the conjugate transpose matrix with inner product

I'm trying to prove that for a certain matrix $A$, and its conjugate transpose $A^*$, we have $⟨Ax,y⟩=⟨x,A^*y⟩$, where $⟨⟩$ represent the inner product. So here it's simply the dot product in $R^n$. ...
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### the complement subspace of a subspace of an ips

$\langle p,q\rangle=\int_{-1}^1p(x)q(x)\,\mathrm{d}x \quad, (p,q\in R_3[x])$\,where $R_3[x]$ is the vector space of all real polynomial of degree less than equal to 3. If $W=\{\,p \in R_3[x]:p(0)=0\}$...
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### Why does the Cauchy-Schwarz Inequality even have a name?

When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of proof. I've always thought the geometric definition ...
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### A conjugate LT in an inner product space

Let $V$ be a finite dimensional complex inner product space. Let $J$ be a conjugate linear map from $V$ to $V$ such that $J^2 =1$. Can we say $\langle Ju, Jv \rangle = \langle v, u \rangle$ for all ...
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### V complex IPS and LT with real matrix entries

V is finite dimensional complex inner product space. There exist a basis such that matrix entries of T in wrt this basis are real. Can we say there will exist an orthonormal basis wrt which matrix ...
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T is a linear operator on V, with an adjoint. V is a complex inner product space, not necessarily of finite dimension. How do I prove $$\left \langle T(x),x \right \rangle=0,\forall x\in V\... 1answer 40 views ### How can there be an inner product space when inner product yields a scalar? [closed] I thought the inner product yields a scalar in both real and complex cases. How can a space be made up of scalars? Taking two vectors a and b, a=(a_1,a_2) and b=(b_1,b_2), the inner product is ... 1answer 74 views ### Prove that R(T^{*})^\perp =N(T) Let V be an inner product space, with T being a linear operator on V. How do I prove that R(T^{*})^\perp =N(T)? I tried setting x\in R(T^{*}) and Ty\in N(T), and set up an inner product = 0 ... 3answers 49 views ### Let V be an inner product space. Prove \langle x, 0\rangle = \langle 0, x\rangle = 0 for x \in V. [closed] Let V be an inner product space. For x \in V, prove \langle x, 0\rangle = \langle 0, x\rangle = 0. 1answer 57 views ### Show that \langle y_i, y_j\rangle = 0 \forall i \neq j. Let y_1, y_2, . . . be a sequence in a Hilbert space. Let V_n be the linear span of \{y_1, y_2, . . . , y_n\}. Assume that for n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥ for each y \in V_n. ... 1answer 73 views ### Finding the orthogonal projection of a given vector on the given subspace W of the inner product space V. Let V = R^3 with the standard inner product u = (2,1,3) and W = \{(x,y,z) : x + 3y - 2z = 0\} I came up with the basis \{(-3,1,0), (2,0,1)\} but these are not orthogonal to each other. I'm ... 1answer 41 views ### Separation of a convex set and a vector by a hyperplane How a hyperplane separates a convex set. Please give me mathematical and geometrical representation and also I want to know the consequences after separations? Actually, I was reading a scholarly ... 2answers 58 views ### If \langle x,y\rangle = \langle x,z\rangle , then y=z PROOF I'm trying to prove a property of inner products. The property is:$$\langle x,y\rangle=\langle x,z\rangle \forall x \in V\Rightarrow y=z$$My proof: Let y=z+w then$$\langle x,z+w\rangle = \...
Q. Let V be a $n-$dimensional space over $\mathbb{C}$ and $T$ a normal operator. Prove that all T-invariant space is $T^{*}$-invariant. A. If $T$ is normal over $\mathbb{C}$ then $V$ admit a basis of ...
Suppose we have a Hilbert space $H$. Is there any explicit expression for the inner product on $H^*$ without resorting to Riesz representation theorem? I am NOT looking for one that uses the ...