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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2
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0answers
250 views

Proof of equivalence in strictly convex spaces

I'm trying to understand a proof and from my current understanding there is one thing not clear/wrong. Maybe you can help: Let $(X, \lVert\cdot\rVert)$ a normed space and $B(0,1)$ the closed unit ...
4
votes
1answer
250 views

Characterization of normed vector spaces of finite dimension

I have this problem: Let $E$ be a normed vector space. $S=\{x\in E : ||x||=1\}$. Show that if $S$ is compact then $\dim E$ is finite. This follows directly from the Riesz's lemma, but in the ...
7
votes
3answers
1k views

Operator norm on product space

I have a bilinear operator $B\colon X \times Y\to Z$ with $X,Y,Z$ normed spaces, and define a norm on $X \times Y$ by $\lVert(x,y)\rVert = \lVert x\rVert_X + \lVert y\rVert_Y$ (using the respective ...
2
votes
2answers
411 views

Point-wise and Norm Convergence of Vectors in a finite dimensional space

I'm trying to prove the theorem, that states, that if I have a normed vector space with a finite dimension (so that each vector I can express as a linear combination $$ ...
18
votes
2answers
4k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
5
votes
1answer
622 views

Different norms on a product space $X \times Y$

It is well known how to define standard product topology on a product space $\prod_{i \in I} X_i$. Assume now that $(X,\lVert \, \cdot \, \rVert_{X})$ and $(Y,\lVert \, \cdot \, \rVert_{Y})$ are ...
4
votes
1answer
232 views

Non-completeness of the space of bounded linear operators

If $X$ and $Y$ are normed spaces I know that the space $B(X,Y)$ of bounded linear functions from $X$ to $Y$, is complete if $Y$ is complete. Is there an example of a pair of normed spaces $X,Y$ s.t. ...