# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

I was asked this in functional analysis class: Let $\mathbb{H}$ be a Hilbert space and we are given an operator T satisfying: $|| Th || < ||h||$ for all $h \in H$. We are asked if ...
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I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality. All the proofs I have referred to ...
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### Frobenius norm of a matrix [closed]

I know that Frobenius norm of a matrix A is equal to the square root of the trace of (A*conjugate transpose(A)). But how do I prove it mathematically?
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### Norm of integral operator in $L^1$

What is the norm of the operator $$T\colon L^1[0,1] \to L^1[0,1]: f\mapsto \left(t\mapsto \int_0^t f(s)ds\right)$$ ?
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### For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
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### Is the max matrix norm induced?

Let $\|A \| = \max_{1 \le i,j \le n} |a_{ij}|$, where $A$ is a square matrix. I can prove that this is a matrix norm, but is it an induced norm?
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### Given a matrix $A$ such that $||A||<1$, prove that $I-A$ is invertible

Note: This is a question seen in class while discussing metric spaces and norms, so my recollection might not be 100% accurate. I saw a proof in class, but I wanted to know if there was a different ...
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### Generalization of $\frac{a + b}{c + d} \leq \text{max}(\frac{a}{c}, \frac{b}{d})$

I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have ...
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### Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_*$ as the norm dual of ...
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### Norm in a dual space

If $f \in X^*$, with $X^*$ the dual space consisting of all linear bounded functionals on a linear normed space $X$. With the norm defined as $||f||_{X^{*}} = \sup_{||x|| \leqslant 1} |f(x)|$. Why ...
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### triangle inequality for a certain norm

Let $d$ be a metric on a (say real) vector space $E$, with the property $$d(x,x+cy)=|c|d(x,x+y)$$ for all $x,y\in E$ and scalars $c$. I am trying to prove that $x\mapsto d(x,0)$ defines a norm. The ...
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### What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?

Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the ...
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### Is closure of convex subset of $X$ is again a convex subset of $X$?
Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ ...