# Tagged Questions

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### Express Norm Using Inner Product

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.
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### Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
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### For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
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### Is any norm induced by some inner product? [duplicate]

It is a well-know fact that an inner product induces some norm. How about the converse? I think it's false but I can't think of an example. I'm thinking some properties like the parallelogram law ...
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### On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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### Topology induced by Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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### Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
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### If every inner product space can be converted into a norm space, then why is there a distinction between the two?

If every inner product space can be converted into a norm space, then why is there a distinction between the two? $$\|x\| = \sqrt{\langle x,x\rangle }$$
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$U,V$ - Euclidean spaces $f:U \rightarrow V$ $f(0)=0$ $\forall _{u,v \in U}:d(f(u),f(v))=d(u,v)$ Prove that $f$ is a linear map. I'm thinking about something like this: $||f(u+v)|| =d(f(u+v),0) = ... 1answer 70 views ### Show that orthogonal complement is trivial I have this subspace of$C[-1,1]$with inner product$\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\}$$ need to prove that$E^\bot=\{0\}$1answer 37 views ### construction of normed linear spaces from inner product spaces How do i construct a normed linear space from a inner product space and verify that what i have suggested is true, that is, a test of verification. Moreover, does the norm have to satisfy the ... 2answers 100 views ### A problem on the bounds of Lp-norms Let$L>0$and$\Omega$be the set of all integrable functions from$[0,L]$to$[0,+\infty]$. Also, Let$f\in \Omega$such that$\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ... 2answers 553 views ### Derivation of the polarization identities? For a real (or complex) inner product space$V$, the inner product can be expressed in terms of the norm as either $$\langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2)$$ or $$\langle ... 2answers 73 views ### Relations between \|x+a\| and \|x-a\| in a normed linear space. 1) Can it happen that \|x+a\|=\|x-a\|=\|x\|+\|a\| when a\ne0? 2) How large can \min(\|x+a\|,\|x-a\|)/\|x\| be when \|x\|\ge \|a\|? (For a inner-product space, the answers are no and ... 1answer 203 views ### Relation between Normed space and inner product Below is what i have proved: If V is a normed vector space over \mathbb{R} satisfies parallelogram equality, then there exists an inner product \langle \bullet,\bullet\rangle such that ... 3answers 444 views ### difference between normed linear space and inner product space I've seen that the definitions of normed linear space and inner product space for a complex vector space V are very close to each other except for the fact that one is defined on V and the other ... 0answers 52 views ### Determining Similarity of Unit Vectors I'm seeking for an injective piecewise continuous function f:\mathbb S^n\rightarrow[0,1] where \mathbb S^N is the set of vectors with L_2 norm equals 1. The piecewise continuity requirement ... 1answer 499 views ### Multiplicative norm on \mathbb{R}[X]. How to prove that : there is no function N\colon \mathbb{R}[X] \rightarrow \mathbb{R}, such that : N is a norm of \mathbb{R}-vector space and N(PQ)=N(P)N(Q) for all P,Q \in \mathbb{R}[X]. ... 1answer 344 views ### Distance between real finite dimensional linear subspaces Is there a usual distance between linear subspaces (V,W) of an n-dimensional normed vector space with inner product? In the case of hyper-planes one could use the angle (based on the inner product ... 1answer 204 views ### If the unit sphere of a normed space is homogeneous is the space an inner product space? Consider a normed vector space V. Suppose that for every pair of unit vectors v,w there exists a linear isometry which sends v to w (and leaves the subspace spanned by v and w invariant). ... 2answers 126 views ### Does (f,Tg)_{L^2} define an inner product space? To my understanding inner product$$(f,g)_{L^2(\mathcal{D})} = \int_\mathcal{D} f(\boldsymbol{x})g(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x},~~\mathcal{D} \subset \mathbb{R}^N$$defines an inner ... 1answer 891 views ### An example of a norm which can't be generated by an inner product I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can ... 1answer 74 views ### Norms on inner product space over \mathbb{R} Definition of the problem Let \left(E,\left\langle \cdot,\cdot\right\rangle \right) be an inner product space over \mathbb{R}. Prove that for all x,y\in E we have$$ \left(\left\Vert ... 1answer 446 views ### Cauchy-Schwarz Inequality proof (for semi-inner-product A-module). I am reading a proof of the following Cauchy-Schwarz Inequality and I don't understand one part of the proof: Theorem: Let$A$be a$C^*$-algebra and let$E$be a semi-inner-product$A$-module. Then ... 1answer 106 views ### Inner product space over$\mathbb{R}$Definition of the problem I have to prove the following statement: Let$\left(E,\left\langle \cdot,\cdot\right\rangle \right)$be an inner product space over$\mathbb{R}$. prove that for all$x,y\in ...
Given a normed vector space $X$. Let $\mathcal{T}$ represent the topology on $X$ induced by the norm. Define: $A$:={topologies that can make the norm continuous}, $B$:={topologies that can make the ...