Let $L,M$, two subspaces of $V$. Prove $L\cup M \ne V$
28819
A union of two subspaces not equal to the vector space.
92774
Prove that a linear subspace of $C([0,1])$ is closed
22295
Subspace of a separable space is separable
13421
Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$
27564
Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
133087
Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
148668
How to show $A\times (B\cup C) = (A\times B)\cup (A\times C)$
33524
Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
74623
Associative rule of subspaces and vector spaces: $(U+V) \cap W = U + (V \cap W)$
91022
How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?
53364
dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$
45337
Suppose that $V, W$ are closed in $X$ and that $V\cap W, V\cup W$ are connected. Show that $V$ and $W$ are connected.
75966
Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements.
60707
Show that if $ \ A \cup B = A$ and $ \ A\cap B = A$ then $ \ A = B$
143979
How can I prove that $|A| + |B| = |A\cup B| + |A\cap B|$?
62065
Prove that $v_1, \dots v_n$ is a basis of V.