# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Intuition of a norm vector space/ infinite dimensional vector space

I'm finding it terribly difficult to build an intuition of what a norm vector space and an infinite dimensional vector space is. There aren't any good notes online that builds the intuition-most ...
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### Examples of infinite dimensional normed vector spaces

In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply ...
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### Need help with this proof, theory of finite change.

Theory: If $f : [a,b] \to X$ is differentiable on (a,b) and continuous on [a,b] in $X$, a normed vector space upon $\langle , \rangle$ then: $$|f(b)-f(a)| \leq \sup_{a <c<b}{\|f'(c)\|}(b-a)$$ ...
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### Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
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### Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
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### The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is ...
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### $A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A$ is non-empty ?
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### Complement of the union of countably many , mutually disjoint , non-empty open balls in $\mathbb R^n , (n >1)$ is path connected?

Let $n \ge 2$ and $\{B_m\}_{m=1}^\infty$ be countably infinitely many , mutually disjoint , non-empty open balls in $\mathbb R^n$ , then is $\mathbb R^n \setminus \cup_{m=1}^\infty B_m$ path-...
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### Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
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### Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
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### Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
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### Image of a function with small BMO norm

This is a question related with the regularity of harmonic maps. Let $N\geq 1$ and $f:\mathbb{R}^N\to \mathbb{S}^2$, where $\mathbb{S}^2=\{x\in \mathbb{R}^3 : \|x\|=1\}$. Assume that the BMO semi-...
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### Show that $\| x-x_j \|_2^2$ is equal to

I have a simple question. $$\|x-c_j\|_2^2 = x^T x-2x^T c_j+c_j^T c_j$$ Where $x,c_j\in\mathbb{R}^n$, $j=1,2,\ldots,n$. I know that $$\|x-c_j\|_2^2=(x-c_1)^2+(x-c_2)^2+\cdots+(x-c_n)^2.$$ But I don't ...
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### When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
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### How to show a norm identity of a weighted sum

I ran across the following identity while reading up on norms. It deals with the square of the $2$-norm of a convex combination. That is, for all $x,y,\in\mathbb{R}^{n}$ and $\rho \in [0,1]$: \begin{...
### Dual norm of $L_p$ space
Given $R^n$ is equipped with the norm $||x|| = (\sum_{k=1}^{n} |x_k|^p)^{\frac{1}{p}}$ for some $p ≥ 1$, what is the induced norm on the conjugate (dual) space? I couldn't figure out how to prove ...