Filter items of a ListBox based on the text of a TextBox using only XAML in WPF

You can use the CollectionViewSource to apply filtering, another example can be found here and here.

Like CodeNaked and devdigital told you CollectionViewSource/CollectionView/ICollectionView are the keys to your goal

It's a MVVM patter but this is a View only related problem so I don't want this code at the ViewModel.

thats not the right way because the View only shows what she get´s but shouldn´t modifi so it should/must be your ViewModel who handel changes

so now some code snips:

    public class myVM
    {
        public CollectionViewSource CollViewSource { get; set; }
        public string SearchFilter
        {
            get;
            set
            {
              if(!string.IsNullOrEmpty(SearchFilter))
                 AddFilter();

                CollViewSource.View.Refresh(); // important to refresh your View
            }
        }
        public myVM(YourCollection)
        {
            CollViewSource = new CollectionViewSource();//onload of your VM class
            CollViewSource.Source = YourCollection;//after ini YourCollection
        }
    }

Xaml Snip:

    <StackPanel>
        <TextBox Height="23" HorizontalAlignment="Left"  Name="tB" VerticalAlignment="Top" 
                 Width="120" Text="{Binding SearchFilter,UpdateSourceTrigger=PropertyChanged}" />
        <DataGrid Name="testgrid" ItemsSource="{Binding CollViewSource.View}"/>
    </StackPanel>

Edit i forgot the Filter

private void AddFilter()
{
    CollViewSource.Filter -= new FilterEventHandler(Filter);
    CollViewSource.Filter += new FilterEventHandler(Filter);  

}

private void Filter(object sender, FilterEventArgs e)
{
    // see Notes on Filter Methods:
    var src = e.Item as YourCollectionItemTyp;
    if (src == null)
        e.Accepted = false;
    else if ( src.FirstName !=null && !src.FirstName.Contains(SearchFilter))// here is FirstName a Property in my YourCollectionItem
        e.Accepted = false;
}

You can do this with a CollectionViewSource. You wouldn't want to do this completely in XAML, as it would be much easier to test this if the filtering code is in your view model (assuming an MVVM design pattern).